Theorem dfrdg4 34180

Description: A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
dfrdg4	⊢ rec(𝐹, 𝐴) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))

Proof of Theorem dfrdg4

Dummy variables 𝑎 𝑏 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrdg3 33678	. 2 ⊢ rec(𝐹, 𝐴) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
2		an12 641	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
3		df-fn 6421	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
4		ancom 460	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (dom 𝑓 = 𝑥 ∧ Fun 𝑓))
5		eqcom 2745	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
6	5	anbi1i 623	. . . . . . . . . 10 ⊢ ((dom 𝑓 = 𝑥 ∧ Fun 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
7	3, 4, 6	3bitri 296	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
8	7	anbi1i 623	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ ((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
9		anass 468	. . . . . . . 8 ⊢ (((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
10	2, 8, 9	3bitri 296	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
11	10	exbii 1851	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
12		vex 3426	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	dmex 7732	. . . . . . 7 ⊢ dom 𝑓 ∈ V
14		eleq1 2826	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
15		raleq 3333	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
16	14, 15	anbi12d 630	. . . . . . . 8 ⊢ (𝑥 = dom 𝑓 → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
17	16	anbi2d 628	. . . . . . 7 ⊢ (𝑥 = dom 𝑓 → ((Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
18	13, 17	ceqsexv 3469	. . . . . 6 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
19	11, 18	bitri 274	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
20		df-rex 3069	. . . . 5 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
21		eldif 3893	. . . . . 6 ⊢ (𝑓 ∈ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
22		elin 3899	. . . . . . . 8 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
23	12	elfuns 34144	. . . . . . . . 9 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
24	12	elima 5963	. . . . . . . . . 10 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
25		df-rex 3069	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓))
26		vex 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27	26, 12	brcnv 5780	. . . . . . . . . . . . . 14 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
28	12, 26	brdomain 34162	. . . . . . . . . . . . . 14 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
29	27, 28	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
30	29	anbi1ci 625	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ (𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
31	30	exbii 1851	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
32	13, 14	ceqsexv 3469	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On) ↔ dom 𝑓 ∈ On)
33	31, 32	bitri 274	. . . . . . . . . 10 ⊢ (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ dom 𝑓 ∈ On)
34	24, 25, 33	3bitri 296	. . . . . . . . 9 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ dom 𝑓 ∈ On)
35	23, 34	anbi12i 626	. . . . . . . 8 ⊢ ((𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
36	22, 35	bitri 274	. . . . . . 7 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
37	36	anbi1i 623	. . . . . 6 ⊢ ((𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
38		brdif 5123	. . . . . . . . . . . . . . 15 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ (𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
39		vex 3426	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
40	12, 39	brco 5768	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
41	28	anbi1i 623	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
42	41	exbii 1851	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
43		breq1 5073	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
44	13, 43	ceqsexv 3469	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
45	42, 44	bitri 274	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
46	13, 39	brcnv 5780	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
47	13	epeli 5488	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
48	46, 47	bitri 274	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
49	40, 45, 48	3bitri 296	. . . . . . . . . . . . . . . 16 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
50	49	anbi1i 623	. . . . . . . . . . . . . . 15 ⊢ ((𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
51	38, 50	bitri 274	. . . . . . . . . . . . . 14 ⊢ (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
52		onelon 6276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53	52	3adant1 1128	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54		brun 5121	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ∨ ⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥))
55		brxp 5627	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}))
56		opelxp 5616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
57	12, 56	mpbiran 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58		velsn 4574	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
59	57, 58	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60		velsn 4574	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ {∪ {𝐴}} ↔ 𝑥 = ∪ {𝐴})
61	59, 60	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}) ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
62	55, 61	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
63		brun 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥))
64	26	brresi 5889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65		opelxp 5616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ Limits ))
66	12, 65	mpbiran 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 ∈ Limits )
67	39	ellimits 34139	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦)
68	66, 67	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ Lim 𝑦)
69		opex 5373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
70	69, 26	brco 5768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥))
71		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝑧 ∈ V
72	12, 39, 71	brimg 34166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟨𝑓, 𝑦⟩Img𝑧 ↔ 𝑧 = (𝑓 “ 𝑦))
73	26	brbigcup 34127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 Bigcup 𝑥 ↔ ∪ 𝑧 = 𝑥)
74	72, 73	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
75	74	exbii 1851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
76	12	imaex 7737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 “ 𝑦) ∈ V
77		unieq 4847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑓 “ 𝑦) → ∪ 𝑧 = ∪ (𝑓 “ 𝑦))
78	77	eqeq1d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑓 “ 𝑦) → (∪ 𝑧 = 𝑥 ↔ ∪ (𝑓 “ 𝑦) = 𝑥))
79	76, 78	ceqsexv 3469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ ∪ (𝑓 “ 𝑦) = 𝑥)
80		eqcom 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∪ (𝑓 “ 𝑦) = 𝑥 ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
81	79, 80	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
82	70, 75, 81	3bitri 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
83	68, 82	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥) ↔ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
84	64, 83	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
85	26	brresi 5889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥))
86		opelxp 5616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
87	12, 86	mpbiran 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
88	39	elrn 5791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
89	71, 39	brsuccf 34170	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧Succ𝑦 ↔ 𝑦 = suc 𝑧)
90	89	exbii 1851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧 𝑧Succ𝑦 ↔ ∃𝑧 𝑦 = suc 𝑧)
91	87, 88, 90	3bitri 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ ∃𝑧 𝑦 = suc 𝑧)
92	69, 26	brco 5768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥))
93		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ 𝑎 ∈ V
94	69, 93	brco 5768	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎))
95	12, 39, 71	brpprod3a 34115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ ∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏))
96		3anrot 1098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
97	93	ideq 5750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑓 I 𝑎 ↔ 𝑓 = 𝑎)
98		equcom 2022	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑓 = 𝑎 ↔ 𝑎 = 𝑓)
99	97, 98	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 I 𝑎 ↔ 𝑎 = 𝑓)
100		vex 3426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ 𝑏 ∈ V
101	100	brbigcup 34127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑦 Bigcup 𝑏 ↔ ∪ 𝑦 = 𝑏)
102		eqcom 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (∪ 𝑦 = 𝑏 ↔ 𝑏 = ∪ 𝑦)
103	101, 102	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑦 Bigcup 𝑏 ↔ 𝑏 = ∪ 𝑦)
104		biid 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
105	99, 103, 104	3anbi123i 1153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
106	96, 105	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
107	106	2exbii 1852	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
108		vuniex 7570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ∪ 𝑦 ∈ V
109		opeq1 4801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110	109	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111		opeq2 4802	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑏 = ∪ 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, ∪ 𝑦⟩)
112	111	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑏 = ∪ 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩))
113	12, 108, 110, 112	ceqsex2v 3473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩)
114	95, 107, 113	3bitri 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩)
115	114	anbi1i 623	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
116	115	exbii 1851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
117		opex 5373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ⟨𝑓, ∪ 𝑦⟩ ∈ V
118		breq1 5073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎))
119	117, 118	ceqsexv 3469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎)
120	12, 108, 93	brapply 34167	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (⟨𝑓, ∪ 𝑦⟩Apply𝑎 ↔ 𝑎 = (𝑓‘∪ 𝑦))
121	119, 120	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ 𝑎 = (𝑓‘∪ 𝑦))
122	94, 116, 121	3bitri 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ 𝑎 = (𝑓‘∪ 𝑦))
123	93, 26	brfullfun 34177	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎FullFun𝐹𝑥 ↔ 𝑥 = (𝐹‘𝑎))
124	122, 123	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
125	124	exbii 1851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
126		fvex 6769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓‘∪ 𝑦) ∈ V
127		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎 = (𝑓‘∪ 𝑦) → (𝐹‘𝑎) = (𝐹‘(𝑓‘∪ 𝑦)))
128	127	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑎 = (𝑓‘∪ 𝑦) → (𝑥 = (𝐹‘𝑎) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
129	126, 128	ceqsexv 3469	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))
130	92, 125, 129	3bitri 296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))
131	91, 130	anbi12i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥) ↔ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
132	85, 131	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
133	84, 132	orbi12i 911	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥) ↔ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
134	63, 133	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
135	62, 134	orbi12i 911	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ∨ ⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥) ↔ ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
136	54, 135	bitri 274	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
137		onzsl 7668	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138		nlim0 6309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ¬ Lim ∅
139		limeq 6263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → (Lim 𝑦 ↔ Lim ∅))
140	138, 139	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → ¬ Lim 𝑦)
141	140	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
142		nsuceq0 6331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ suc 𝑧 ≠ ∅
143		neeq2 3006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ∅ → (suc 𝑧 ≠ 𝑦 ↔ suc 𝑧 ≠ ∅))
144	142, 143	mpbiri 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = ∅ → suc 𝑧 ≠ 𝑦)
145	144	necomd 2998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146	145	neneqd 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147	146	nexdv 1940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148	147	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
149		ioran 980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) ↔ (¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∧ ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
150	141, 148, 149	sylanbrc 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → ¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
151		orel2 887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
152	150, 151	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
153		iftrue 4462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154		unisnif 34154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155	153, 154	eqtr4di 2797	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ {𝐴})
156	155	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = ∪ {𝐴}))
157	156	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → (𝑥 = ∪ {𝐴} → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
158	157	adantld 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ∅ → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
159	152, 158	syld 47	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
160	156	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → 𝑥 = ∪ {𝐴}))
161	160	anc2li 555	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
162		orc 863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
163	161, 162	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
164	159, 163	impbid 211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
165		neeq1 3005	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166	142, 165	mpbiri 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → 𝑦 ≠ ∅)
167	166	neneqd 2947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168	167	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
169	168	rexlimivw 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
170		orel1 885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
171	169, 170	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
172		nlimsucg 7664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
173	172	elv 3428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ¬ Lim suc 𝑧
174		limeq 6263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175	173, 174	mtbiri 326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176	175	rexlimivw 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
177	176	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
178		orel1 885	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
179	177, 178	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
180	142	neii 2944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ¬ suc 𝑧 = ∅
181	180	iffalsei 4466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))
182		iffalse 4465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (¬ Lim suc 𝑧 → if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))) = (𝐹‘(𝑓‘∪ suc 𝑧)))
183	71, 172, 182	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))) = (𝐹‘(𝑓‘∪ suc 𝑧))
184	181, 183	eqtri 2766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
185		eqeq1 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186		unieq 4847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
187	186	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
188	187	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
189	174, 188	ifbieq2d 4482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = suc 𝑧 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))))
190	185, 189	ifbieq2d 4482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))))
191	184, 190, 188	3eqtr4a 2805	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
192	191	rexlimivw 3210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
193	192	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
194	193	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = (𝐹‘(𝑓‘∪ 𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
195	194	adantld 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
196	171, 179, 195	3syld 60	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
197		rexex 3167	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ∃𝑧 𝑦 = suc 𝑧)
198	193	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
199		olc 864	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
200	199	olcd 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
201	197, 198, 200	syl6an 680	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
202	196, 201	impbid 211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
203	140	con2i 139	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Lim 𝑦 → ¬ 𝑦 = ∅)
204	203	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Lim 𝑦 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
205	204, 170	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
206	175	exlimiv 1934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∃𝑧 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
207	206	con2i 139	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Lim 𝑦 → ¬ ∃𝑧 𝑦 = suc 𝑧)
208	207	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Lim 𝑦 → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
209		orel2 887	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦))))
210	208, 209	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Lim 𝑦 → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦))))
211	203	iffalsed 4467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))
212		iftrue 4462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
213	211, 212	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ (𝑓 “ 𝑦))
214	213	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = ∪ (𝑓 “ 𝑦)))
215	214	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Lim 𝑦 → (𝑥 = ∪ (𝑓 “ 𝑦) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
216	215	adantld 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Lim 𝑦 → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
217	205, 210, 216	3syld 60	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
218	217	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
219	214	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → 𝑥 = ∪ (𝑓 “ 𝑦)))
220	219	anc2li 555	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦))))
221		orc 863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
222	221	olcd 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
223	220, 222	syl6 35	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
224	223	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ V ∧ Lim 𝑦) → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
225	218, 224	impbid 211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
226	164, 202, 225	3jaoi 1425	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
227	137, 226	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ On → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
228	136, 227	syl5bb 282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ On → (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
229	53, 228	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
230	26, 69	brcnv 5780	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
231	12, 39, 26	brapply 34167	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
232	230, 231	bitri 274	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦))
233	232	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦)))
234	229, 233	anbi12d 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → ((⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ∧ 𝑥 = (𝑓‘𝑦))))
235	234	biancomd 463	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → ((⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓‘𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
236	235	exbidv 1925	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
237		df-br 5071	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))
238	69	elfix 34132	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))) ↔ ⟨𝑓, 𝑦⟩(◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))⟨𝑓, 𝑦⟩)
239	69, 69	brco 5768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑓, 𝑦⟩(◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩))
240	237, 238, 239	3bitri 296	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩))
241		fvex 6769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓‘𝑦) ∈ V
242	241	eqvinc 3571	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
243	236, 240, 242	3bitr4g 313	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
244	243	notbid 317	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
245	244	3expia 1119	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (𝑦 ∈ dom 𝑓 → (¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
246	245	pm5.32d 576	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → ((𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
247		annim 403	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
248	246, 247	bitrdi 286	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → ((𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
249	51, 248	syl5bb 282	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
250	249	exbidv 1925	. . . . . . . . . . . 12 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ ∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
251		exnal 1830	. . . . . . . . . . . 12 ⊢ (∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
252	250, 251	bitr2di 287	. . . . . . . . . . 11 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦))
253	12	eldm 5798	. . . . . . . . . . 11 ⊢ (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦)
254	252, 253	bitr4di 288	. . . . . . . . . 10 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
255	254	con1bid 355	. . . . . . . . 9 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
256		df-ral 3068	. . . . . . . . 9 ⊢ (∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
257	255, 256	bitr4di 288	. . . . . . . 8 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
258	257	pm5.32i 574	. . . . . . 7 ⊢ (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
259		anass 468	. . . . . . 7 ⊢ (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
260	258, 259	bitri 274	. . . . . 6 ⊢ (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
261	21, 37, 260	3bitri 296	. . . . 5 ⊢ (𝑓 ∈ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
262	19, 20, 261	3bitr4ri 303	. . . 4 ⊢ (𝑓 ∈ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
263	262	abbi2i 2878	. . 3 ⊢ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
264	263	unieqi 4849	. 2 ⊢ ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
265	1, 264	eqtr4i 2769	1 ⊢ rec(𝐹, 𝐴) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882  ∅c0 4253  ifcif 4456  {csn 4558  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070   I cid 5479   E cep 5485   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   “ cima 5583   ∘ ccom 5584  Oncon0 6251  Lim wlim 6252  suc csuc 6253  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  reccrdg 8211  pprodcpprod 34060   Bigcup cbigcup 34063   Fix cfix 34064   Limits climits 34065   Funs cfuns 34066  Imgcimg 34071  Domaincdomain 34072  Applycapply 34074  Succcsuccf 34077  FullFuncfullfn 34079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-symdif 4173  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-txp 34083  df-pprod 34084  df-bigcup 34087  df-fix 34088  df-limits 34089  df-funs 34090  df-singleton 34091  df-singles 34092  df-image 34093  df-cart 34094  df-img 34095  df-domain 34096  df-cup 34098  df-succf 34101  df-apply 34102  df-funpart 34103  df-fullfun 34104

This theorem is referenced by: (None)