Theorem dfrdg4 36145

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 35988	. 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 645	. . . . . . . 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 6495	. . . . . . . . . 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 2743	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
6	5	anbi1i 624	. . . . . . . . . 10 ⊢ ((dom 𝑓 = 𝑥 ∧ Fun 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
7	3, 4, 6	3bitri 297	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
8	7	anbi1i 624	. . . . . . . 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 297	. . . . . . 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 1849	. . . . . 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 3444	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	dmex 7851	. . . . . . 7 ⊢ dom 𝑓 ∈ V
14		eleq1 2824	. . . . . . . . 9 ⊢ (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
15		raleq 3293	. . . . . . . . 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 632	. . . . . . . 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 630	. . . . . . 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 3490	. . . . . 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 275	. . . . 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 3061	. . . . 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 3911	. . . . . 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 3917	. . . . . . . 8 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
23	12	elfuns 36107	. . . . . . . . 9 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
24	12	elima 6024	. . . . . . . . . 10 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
25		df-rex 3061	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓))
26		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27	26, 12	brcnv 5831	. . . . . . . . . . . . . 14 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
28	12, 26	brdomain 36125	. . . . . . . . . . . . . 14 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
29	27, 28	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
30	29	anbi1ci 626	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ (𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
31	30	exbii 1849	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
32	13, 14	ceqsexv 3490	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On) ↔ dom 𝑓 ∈ On)
33	31, 32	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ dom 𝑓 ∈ On)
34	24, 25, 33	3bitri 297	. . . . . . . . 9 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ dom 𝑓 ∈ On)
35	23, 34	anbi12i 628	. . . . . . . 8 ⊢ ((𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
36	22, 35	bitri 275	. . . . . . 7 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
37	36	anbi1i 624	. . . . . 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 5151	. . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
40	12, 39	brco 5819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
41	28	anbi1i 624	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
42	41	exbii 1849	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
43		breq1 5101	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
44	13, 43	ceqsexv 3490	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
45	42, 44	bitri 275	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
46	13, 39	brcnv 5831	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
47	13	epeli 5526	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
48	46, 47	bitri 275	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
49	40, 45, 48	3bitri 297	. . . . . . . . . . . . . . . 16 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
50	49	anbi1i 624	. . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . 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 6342	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53	52	3adant1 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54		brun 5149	. . . . . . . . . . . . . . . . . . . . . . . . 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 5673	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}))
56		opelxp 5660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
57	12, 56	mpbiran 709	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58		velsn 4596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
59	57, 58	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60		velsn 4596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ {∪ {𝐴}} ↔ 𝑥 = ∪ {𝐴})
61	59, 60	anbi12i 628	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}) ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
62	55, 61	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
63		brun 5149	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65		opelxp 5660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ Limits ))
66	12, 65	mpbiran 709	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 ∈ Limits )
67	39	ellimits 36102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ Limits ↔ Lim 𝑦)
68	66, 67	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ Lim 𝑦)
69		opex 5412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ⟨𝑓, 𝑦⟩ ∈ V
70	69, 26	brco 5819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥))
71		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝑧 ∈ V
72	12, 39, 71	brimg 36129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟨𝑓, 𝑦⟩Img𝑧 ↔ 𝑧 = (𝑓 “ 𝑦))
73	26	brbigcup 36090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 Bigcup 𝑥 ↔ ∪ 𝑧 = 𝑥)
74	72, 73	anbi12i 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
75	74	exbii 1849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
76	12	imaex 7856	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 “ 𝑦) ∈ V
77		unieq 4874	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑓 “ 𝑦) → ∪ 𝑧 = ∪ (𝑓 “ 𝑦))
78	77	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑓 “ 𝑦) → (∪ 𝑧 = 𝑥 ↔ ∪ (𝑓 “ 𝑦) = 𝑥))
79	76, 78	ceqsexv 3490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ ∪ (𝑓 “ 𝑦) = 𝑥)
80		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∪ (𝑓 “ 𝑦) = 𝑥 ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
81	79, 80	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
82	70, 75, 81	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
83	68, 82	anbi12i 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥) ↔ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
84	64, 83	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
85	26	brresi 5947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5660	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
87	12, 86	mpbiran 709	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
88	39	elrn 5842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
89	71, 39	brsuccf 36134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧Succ𝑦 ↔ 𝑦 = suc 𝑧)
90	89	exbii 1849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧 𝑧Succ𝑦 ↔ ∃𝑧 𝑦 = suc 𝑧)
91	87, 88, 90	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ ∃𝑧 𝑦 = suc 𝑧)
92	69, 26	brco 5819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥))
93		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ 𝑎 ∈ V
94	69, 93	brco 5819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎))
95	12, 39, 71	brpprod3a 36078	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ ∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏))
96		3anrot 1099	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
97	93	ideq 5801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑓 I 𝑎 ↔ 𝑓 = 𝑎)
98		equcom 2019	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑓 = 𝑎 ↔ 𝑎 = 𝑓)
99	97, 98	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 I 𝑎 ↔ 𝑎 = 𝑓)
100		vex 3444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ 𝑏 ∈ V
101	100	brbigcup 36090	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑦 Bigcup 𝑏 ↔ ∪ 𝑦 = 𝑏)
102		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (∪ 𝑦 = 𝑏 ↔ 𝑏 = ∪ 𝑦)
103	101, 102	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑦 Bigcup 𝑏 ↔ 𝑏 = ∪ 𝑦)
104		biid 261	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
105	99, 103, 104	3anbi123i 1155	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
106	96, 105	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
107	106	2exbii 1850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
108		vuniex 7684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ∪ 𝑦 ∈ V
109		opeq1 4829	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110	109	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111		opeq2 4830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑏 = ∪ 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, ∪ 𝑦⟩)
112	111	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑏 = ∪ 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩))
113	12, 108, 110, 112	ceqsex2v 3494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩)
114	95, 107, 113	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩)
115	114	anbi1i 624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
116	115	exbii 1849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
117		opex 5412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ⟨𝑓, ∪ 𝑦⟩ ∈ V
118		breq1 5101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎))
119	117, 118	ceqsexv 3490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎)
120	12, 108, 93	brapply 36130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (⟨𝑓, ∪ 𝑦⟩Apply𝑎 ↔ 𝑎 = (𝑓‘∪ 𝑦))
121	119, 120	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ 𝑎 = (𝑓‘∪ 𝑦))
122	94, 116, 121	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ 𝑎 = (𝑓‘∪ 𝑦))
123	93, 26	brfullfun 36142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎FullFun𝐹𝑥 ↔ 𝑥 = (𝐹‘𝑎))
124	122, 123	anbi12i 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
125	124	exbii 1849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
126		fvex 6847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓‘∪ 𝑦) ∈ V
127		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎 = (𝑓‘∪ 𝑦) → (𝐹‘𝑎) = (𝐹‘(𝑓‘∪ 𝑦)))
128	127	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑎 = (𝑓‘∪ 𝑦) → (𝑥 = (𝐹‘𝑎) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
129	126, 128	ceqsexv 3490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))
130	92, 125, 129	3bitri 297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))
131	91, 130	anbi12i 628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
133	84, 132	orbi12i 914	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . . . . . . . . . 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 914	. . . . . . . . . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . . . . . . . . . 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 7788	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138		nlim0 6377	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ¬ Lim ∅
139		limeq 6329	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → (Lim 𝑦 ↔ Lim ∅))
140	138, 139	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → ¬ Lim 𝑦)
141	140	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
142		nsuceq0 6402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ suc 𝑧 ≠ ∅
143		neeq2 2995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ∅ → (suc 𝑧 ≠ 𝑦 ↔ suc 𝑧 ≠ ∅))
144	142, 143	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = ∅ → suc 𝑧 ≠ 𝑦)
145	144	necomd 2987	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146	145	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147	146	nexdv 1937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148	147	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
149		ioran 985	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → ¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
151		orel2 890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154		unisnif 36117	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155	153, 154	eqtr4di 2789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ {𝐴})
156	155	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = ∪ {𝐴}))
157	156	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 867	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 212	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166	142, 165	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → 𝑦 ≠ ∅)
167	166	neneqd 2937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168	167	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
169	168	rexlimivw 3133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
170		orel1 888	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7784	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
173	172	elv 3445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ¬ Lim suc 𝑧
174		limeq 6329	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175	173, 174	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176	175	rexlimivw 3133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 888	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ¬ suc 𝑧 = ∅
181	180	iffalsei 4489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
185		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186		unieq 4874	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
187	186	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
188	187	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
189	174, 188	ifbieq2d 4506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
192	191	rexlimivw 3133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3066	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ∃𝑧 𝑦 = suc 𝑧)
198	193	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 874	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 212	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1931	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
213	211, 212	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ (𝑓 “ 𝑦))
214	213	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = ∪ (𝑓 “ 𝑦)))
215	214	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 867	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
222	221	olcd 874	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 212	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1430	. . . . . . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . . . . . 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	bitrid 283	. . . . . . . . . . . . . . . . . . . . . . 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 5831	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
231	12, 39, 26	brapply 36130	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
232	230, 231	bitri 275	. . . . . . . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . . . . . . 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 1922	. . . . . . . . . . . . . . . . . . 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 5099	. . . . . . . . . . . . . . . . . . . 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 36095	. . . . . . . . . . . . . . . . . . . 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 5819	. . . . . . . . . . . . . . . . . . . 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 297	. . . . . . . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓‘𝑦) ∈ V
242	241	eqvinc 3603	. . . . . . . . . . . . . . . . . . 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 314	. . . . . . . . . . . . . . . . . 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 318	. . . . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . . . . 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 577	. . . . . . . . . . . . . . 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 287	. . . . . . . . . . . . . 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	bitrid 283	. . . . . . . . . . . . 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 1922	. . . . . . . . . . . 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 1828	. . . . . . . . . . . 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 288	. . . . . . . . . . 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 5849	. . . . . . . . . . 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 289	. . . . . . . . . 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 3052	. . . . . . . . 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 289	. . . . . . . 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 275	. . . . . 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 297	. . . . 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 304	. . . 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	eqabi 2871	. . 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 4875	. 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 2762	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 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∖ cdif 3898   ∪ cun 3899   ∩ cin 3900  ∅c0 4285  ifcif 4479  {csn 4580  ⟨cop 4586  ∪ cuni 4863   class class class wbr 5098   I cid 5518   E cep 5523   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   “ cima 5627   ∘ ccom 5628  Oncon0 6317  Lim wlim 6318  suc csuc 6319  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  reccrdg 8340  pprodcpprod 36023   Bigcup cbigcup 36026   Fix cfix 36027   Limits climits 36028   Funs cfuns 36029  Imgcimg 36034  Domaincdomain 36035  Applycapply 36037  Succcsuccf 36040  FullFuncfullfn 36042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-symdif 4205  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-txp 36046  df-pprod 36047  df-bigcup 36050  df-fix 36051  df-limits 36052  df-funs 36053  df-singleton 36054  df-singles 36055  df-image 36056  df-cart 36057  df-img 36058  df-domain 36059  df-cup 36061  df-succf 36064  df-apply 36065  df-funpart 36066  df-fullfun 36067

This theorem is referenced by: (None)