Theorem dfrdg4 36152

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 35995	. 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 646	. . . . . . . 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 6496	. . . . . . . . . 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 2744	. . . . . . . . . . 11 ⊢ (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
6	5	anbi1i 625	. . . . . . . . . 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 625	. . . . . . . 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 1850	. . . . . 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 3434	. . . . . . . 8 ⊢ 𝑓 ∈ V
13	12	dmex 7854	. . . . . . 7 ⊢ dom 𝑓 ∈ V
14		eleq1 2825	. . . . . . . . 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 633	. . . . . . . 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 631	. . . . . . 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 3479	. . . . . 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 3063	. . . . 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 3900	. . . . . 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 3906	. . . . . . . 8 ⊢ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
23	12	elfuns 36114	. . . . . . . . 9 ⊢ (𝑓 ∈ Funs ↔ Fun 𝑓)
24	12	elima 6025	. . . . . . . . . 10 ⊢ (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
25		df-rex 3063	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓))
26		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
27	26, 12	brcnv 5832	. . . . . . . . . . . . . 14 ⊢ (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
28	12, 26	brdomain 36132	. . . . . . . . . . . . . 14 ⊢ (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
29	27, 28	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
30	29	anbi1ci 627	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ (𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
31	30	exbii 1850	. . . . . . . . . . 11 ⊢ (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
32	13, 14	ceqsexv 3479	. . . . . . . . . . 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 629	. . . . . . . 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 625	. . . . . 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 5139	. . . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
40	12, 39	brco 5820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
41	28	anbi1i 625	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
42	41	exbii 1850	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
43		breq1 5089	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
44	13, 43	ceqsexv 3479	. . . . . . . . . . . . . . . . . 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 5832	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
47	13	epeli 5527	. . . . . . . . . . . . . . . . . 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 625	. . . . . . . . . . . . . . 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 6343	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53	52	3adant1 1131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54		brun 5137	. . . . . . . . . . . . . . . . . . . . . . . . 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 5674	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}))
56		opelxp 5661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
57	12, 56	mpbiran 710	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58		velsn 4584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
59	57, 58	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60		velsn 4584	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ {∪ {𝐴}} ↔ 𝑥 = ∪ {𝐴})
61	59, 60	anbi12i 629	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5137	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 5948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65		opelxp 5661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ Limits ))
66	12, 65	mpbiran 710	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 ∈ Limits )
67	39	ellimits 36109	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥))
71		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝑧 ∈ V
72	12, 39, 71	brimg 36136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (⟨𝑓, 𝑦⟩Img𝑧 ↔ 𝑧 = (𝑓 “ 𝑦))
73	26	brbigcup 36097	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 Bigcup 𝑥 ↔ ∪ 𝑧 = 𝑥)
74	72, 73	anbi12i 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
75	74	exbii 1850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
76	12	imaex 7859	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑓 “ 𝑦) ∈ V
77		unieq 4862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑧 = (𝑓 “ 𝑦) → ∪ 𝑧 = ∪ (𝑓 “ 𝑦))
78	77	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 = (𝑓 “ 𝑦) → (∪ 𝑧 = 𝑥 ↔ ∪ (𝑓 “ 𝑦) = 𝑥))
79	76, 78	ceqsexv 3479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ ∪ (𝑓 “ 𝑦) = 𝑥)
80		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
87	12, 86	mpbiran 710	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
88	39	elrn 5843	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
89	71, 39	brsuccf 36141	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑧Succ𝑦 ↔ 𝑦 = suc 𝑧)
90	89	exbii 1850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥))
93		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ 𝑎 ∈ V
94	69, 93	brco 5820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎))
95	12, 39, 71	brpprod3a 36085	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ ∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏))
96		3anrot 1100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
97	93	ideq 5802	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑓 I 𝑎 ↔ 𝑓 = 𝑎)
98		equcom 2020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑓 = 𝑎 ↔ 𝑎 = 𝑓)
99	97, 98	bitri 275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑓 I 𝑎 ↔ 𝑎 = 𝑓)
100		vex 3434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ 𝑏 ∈ V
101	100	brbigcup 36097	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (𝑦 Bigcup 𝑏 ↔ ∪ 𝑦 = 𝑏)
102		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
108		vuniex 7687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ∪ 𝑦 ∈ V
109		opeq1 4817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110	109	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111		opeq2 4818	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑏 = ∪ 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, ∪ 𝑦⟩)
112	111	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (𝑏 = ∪ 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩))
113	12, 108, 110, 112	ceqsex2v 3483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
116	115	exbii 1850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
117		opex 5412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ⟨𝑓, ∪ 𝑦⟩ ∈ V
118		breq1 5089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎))
119	117, 118	ceqsexv 3479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎)
120	12, 108, 93	brapply 36137	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 36149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎FullFun𝐹𝑥 ↔ 𝑥 = (𝐹‘𝑎))
124	122, 123	anbi12i 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
125	124	exbii 1850	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
126		fvex 6848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑓‘∪ 𝑦) ∈ V
127		fveq2 6835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑎 = (𝑓‘∪ 𝑦) → (𝐹‘𝑎) = (𝐹‘(𝑓‘∪ 𝑦)))
128	127	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑎 = (𝑓‘∪ 𝑦) → (𝑥 = (𝐹‘𝑎) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
129	126, 128	ceqsexv 3479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 915	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 915	. . . . . . . . . . . . . . . . . . . . . . . . 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 7791	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138		nlim0 6378	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ¬ Lim ∅
139		limeq 6330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ suc 𝑧 ≠ ∅
143		neeq2 2996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = ∅ → (suc 𝑧 ≠ 𝑦 ↔ suc 𝑧 ≠ ∅))
144	142, 143	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = ∅ → suc 𝑧 ≠ 𝑦)
145	144	necomd 2988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146	145	neneqd 2938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147	146	nexdv 1938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148	147	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = ∅ → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
149		ioran 986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 = ∅ → ¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
151		orel2 891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154		unisnif 36124	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155	153, 154	eqtr4di 2790	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ {𝐴})
156	155	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 868	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2995	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166	142, 165	mpbiri 258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → 𝑦 ≠ ∅)
167	166	neneqd 2938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168	167	intnanrd 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
169	168	rexlimivw 3135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
170		orel1 889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
173	172	elv 3435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ¬ Lim suc 𝑧
174		limeq 6330	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175	173, 174	mtbiri 327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176	175	rexlimivw 3135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ¬ suc 𝑧 = ∅
181	180	iffalsei 4477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
185		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186		unieq 4862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
187	186	fveq2d 6839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
188	187	fveq2d 6839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
189	174, 188	ifbieq2d 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
192	191	rexlimivw 3135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3068	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 875	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 685	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4478	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
213	211, 212	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ (𝑓 “ 𝑦))
214	213	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
222	221	olcd 875	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1431	. . . . . . . . . . . . . . . . . . . . . . . . 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 5832	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
231	12, 39, 26	brapply 36137	. . . . . . . . . . . . . . . . . . . . . . . 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 633	. . . . . . . . . . . . . . . . . . . . 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 1923	. . . . . . . . . . . . . . . . . . 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 5087	. . . . . . . . . . . . . . . . . . . 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 36102	. . . . . . . . . . . . . . . . . . . 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 5820	. . . . . . . . . . . . . . . . . . . 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 6848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓‘𝑦) ∈ V
242	241	eqvinc 3592	. . . . . . . . . . . . . . . . . . 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 1122	. . . . . . . . . . . . . . . 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 1923	. . . . . . . . . . . 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 1829	. . . . . . . . . . . 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 5850	. . . . . . . . . . 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 3053	. . . . . . . . 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 2872	. . 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 4863	. 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 2763	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 848   ∨ w3o 1086   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889  ∅c0 4274  ifcif 4467  {csn 4568  ⟨cop 4574  ∪ cuni 4851   class class class wbr 5086   I cid 5519   E cep 5524   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628   ∘ ccom 5629  Oncon0 6318  Lim wlim 6319  suc csuc 6320  Fun wfun 6487   Fn wfn 6488  ‘cfv 6493  reccrdg 8342  pprodcpprod 36030   Bigcup cbigcup 36033   Fix cfix 36034   Limits climits 36035   Funs cfuns 36036  Imgcimg 36041  Domaincdomain 36042  Applycapply 36044  Succcsuccf 36047  FullFuncfullfn 36049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-symdif 4194  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-txp 36053  df-pprod 36054  df-bigcup 36057  df-fix 36058  df-limits 36059  df-funs 36060  df-singleton 36061  df-singles 36062  df-image 36063  df-cart 36064  df-img 36065  df-domain 36066  df-cup 36068  df-succf 36071  df-apply 36072  df-funpart 36073  df-fullfun 36074

This theorem is referenced by: (None)