Theorem dfrdg2 35860

Description: Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
dfrdg2	⊢ (𝐼 ∈ 𝑉 → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})

Distinct variable groups:   𝑓,𝐹,𝑥,𝑦   𝑓,𝐼,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑓)

Proof of Theorem dfrdg2

Dummy variables 𝑔 𝑖 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rdgeq2 8339	. . 3 ⊢ (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2		ifeq1 4480	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
3	2	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → ((𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
4	3	ralbidv 3156	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
5	4	anbi2d 630	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
6	5	rexbidv 3157	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
7	6	abbidv 2799	. . . 4 ⊢ (𝑖 = 𝐼 → {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
8	7	unieqd 4873	. . 3 ⊢ (𝑖 = 𝐼 → ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
9	1, 8	eqeq12d 2749	. 2 ⊢ (𝑖 = 𝐼 → (rec(𝐹, 𝑖) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))} ↔ rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}))
10		df-rdg 8337	. . 3 ⊢ rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
11		dfrecs3 8300	. . 3 ⊢ recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)))}
12		vex 3441	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
13	12	resex 5984	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ 𝑦) ∈ V
14		eqeq1 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔 = ∅ ↔ (𝑓 ↾ 𝑦) = ∅))
15		relres 5960	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑓 ↾ 𝑦)
16		reldm0 5874	. . . . . . . . . . . . . . . 16 ⊢ (Rel (𝑓 ↾ 𝑦) → ((𝑓 ↾ 𝑦) = ∅ ↔ dom (𝑓 ↾ 𝑦) = ∅))
17	15, 16	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ↾ 𝑦) = ∅ ↔ dom (𝑓 ↾ 𝑦) = ∅)
18	14, 17	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔 = ∅ ↔ dom (𝑓 ↾ 𝑦) = ∅))
19		dmeq 5849	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → dom 𝑔 = dom (𝑓 ↾ 𝑦))
20		limeq 6325	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑔 = dom (𝑓 ↾ 𝑦) → (Lim dom 𝑔 ↔ Lim dom (𝑓 ↾ 𝑦)))
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (Lim dom 𝑔 ↔ Lim dom (𝑓 ↾ 𝑦)))
22		rneq 5882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = ran (𝑓 ↾ 𝑦))
23		df-ima 5634	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
24	22, 23	eqtr4di 2786	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = (𝑓 “ 𝑦))
25	24	unieqd 4873	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ ran 𝑔 = ∪ (𝑓 “ 𝑦))
26		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → 𝑔 = (𝑓 ↾ 𝑦))
27	19	unieqd 4873	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ dom 𝑔 = ∪ dom (𝑓 ↾ 𝑦))
28	26, 27	fveq12d 6837	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔‘∪ dom 𝑔) = ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))
29	28	fveq2d 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))
30	21, 25, 29	ifbieq12d 4505	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))))
31	18, 30	ifbieq2d 4503	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))))
32		eqid 2733	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
33		vex 3441	. . . . . . . . . . . . . 14 ⊢ 𝑖 ∈ V
34		imaexg 7851	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑦) ∈ V)
35	12, 34	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) ∈ V
36	35	uniex 7682	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝑓 “ 𝑦) ∈ V
37		fvex 6843	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) ∈ V
38	36, 37	ifex 4527	. . . . . . . . . . . . . 14 ⊢ if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))) ∈ V
39	33, 38	ifex 4527	. . . . . . . . . . . . 13 ⊢ if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))) ∈ V
40	31, 32, 39	fvmpt 6937	. . . . . . . . . . . 12 ⊢ ((𝑓 ↾ 𝑦) ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)) = if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))))
41	13, 40	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)) = if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))))
42		dmres 5967	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ↾ 𝑦) = (𝑦 ∩ dom 𝑓)
43		onelss 6355	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
44	43	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
45	44	3adant2 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
46		fndm 6591	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47	46	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom 𝑓 = 𝑥)
48	45, 47	sseqtrrd 3968	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ dom 𝑓)
49		dfss2 3916	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ dom 𝑓 ↔ (𝑦 ∩ dom 𝑓) = 𝑦)
50	48, 49	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ dom 𝑓) = 𝑦)
51	42, 50	eqtrid 2780	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom (𝑓 ↾ 𝑦) = 𝑦)
52		eqeq1 2737	. . . . . . . . . . . . . 14 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (dom (𝑓 ↾ 𝑦) = ∅ ↔ 𝑦 = ∅))
53		limeq 6325	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (Lim dom (𝑓 ↾ 𝑦) ↔ Lim 𝑦))
54		unieq 4871	. . . . . . . . . . . . . . . . 17 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ∪ dom (𝑓 ↾ 𝑦) = ∪ 𝑦)
55	54	fveq2d 6834	. . . . . . . . . . . . . . . 16 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)) = ((𝑓 ↾ 𝑦)‘∪ 𝑦))
56	55	fveq2d 6834	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))
57	53, 56	ifbieq2d 4503	. . . . . . . . . . . . . 14 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))) = if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))))
58	52, 57	ifbieq2d 4503	. . . . . . . . . . . . 13 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))))
59		onelon 6338	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
60		eloni 6323	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → Ord 𝑦)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
62	61	3adant2 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
63		ordzsl 7783	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64		iftrue 4482	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = 𝑖)
65		iftrue 4482	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = 𝑖)
66	64, 65	eqtr4d 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
67		vex 3441	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
68	67	sucid 6397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ suc 𝑧
69		fvres 6849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ suc 𝑧 → ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓‘𝑧))
70	68, 69	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓‘𝑧)
71		eloni 6323	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ On → Ord 𝑧)
72		ordunisuc 7770	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑧 → ∪ suc 𝑧 = 𝑧)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ On → ∪ suc 𝑧 = 𝑧)
74	73	fveq2d 6834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
75	73	fveq2d 6834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → (𝑓‘∪ suc 𝑧) = (𝑓‘𝑧))
76	70, 74, 75	3eqtr4a 2794	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = (𝑓‘∪ suc 𝑧))
77	76	fveq2d 6834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
78		nsuceq0 6398	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑧 ≠ ∅
79	78	neii 2931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ suc 𝑧 = ∅
80	79	iffalsei 4486	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))
81		nlimsucg 7780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
82		iffalse 4485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ Lim suc 𝑧 → if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))
83	67, 81, 82	mp2b 10	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))
84	80, 83	eqtri 2756	. . . . . . . . . . . . . . . . . . 19 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))
85	79	iffalsei 4486	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))
86		iffalse 4485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ Lim suc 𝑧 → if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))) = (𝐹‘(𝑓‘∪ suc 𝑧)))
87	67, 81, 86	mp2b 10	. . . . . . . . . . . . . . . . . . . 20 ⊢ if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))) = (𝐹‘(𝑓‘∪ suc 𝑧))
88	85, 87	eqtri 2756	. . . . . . . . . . . . . . . . . . 19 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
89	77, 84, 88	3eqtr4g 2793	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ On → if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))))
90		eqeq1 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91		limeq 6325	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92		reseq2 5929	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → (𝑓 ↾ 𝑦) = (𝑓 ↾ suc 𝑧))
93		unieq 4871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
94	92, 93	fveq12d 6837	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → ((𝑓 ↾ 𝑦)‘∪ 𝑦) = ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))
95	94	fveq2d 6834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))
96	91, 95	ifbieq2d 4503	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = suc 𝑧 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))))
97	90, 96	ifbieq2d 4503	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))))
98	93	fveq2d 6834	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
99	98	fveq2d 6834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
100	91, 99	ifbieq2d 4503	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = suc 𝑧 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))))
101	90, 100	ifbieq2d 4503	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))))
102	97, 101	eqeq12d 2749	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = suc 𝑧 → (if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))))))
103	89, 102	syl5ibrcom 247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ On → (𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
104	103	rexlimiv 3127	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
105		iftrue 4482	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
106		df-lim 6318	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim 𝑦 ↔ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦))
107	106	simp2bi 1146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝑦 → 𝑦 ≠ ∅)
108	107	neneqd 2934	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑦 → ¬ 𝑦 = ∅)
109	108	iffalsed 4487	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))))
110		iftrue 4482	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
111	105, 109, 110	3eqtr4d 2778	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))
112	108	iffalsed 4487	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))
113	111, 112	eqtr4d 2771	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
114	66, 104, 113	3jaoi 1430	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦) → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
115	63, 114	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
116	62, 115	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
117	58, 116	sylan9eqr 2790	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) ∧ dom (𝑓 ↾ 𝑦) = 𝑦) → if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
118	51, 117	mpdan 687	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → if(dom (𝑓 ↾ 𝑦) = ∅, 𝑖, if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
119	41, 118	eqtrid 2780	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))
120	119	eqeq2d 2744	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)) ↔ (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
121	120	3expa 1118	. . . . . . . 8 ⊢ (((𝑥 ∈ On ∧ 𝑓 Fn 𝑥) ∧ 𝑦 ∈ 𝑥) → ((𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)) ↔ (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
122	121	ralbidva 3154	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
123	122	pm5.32da 579	. . . . . 6 ⊢ (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦))) ↔ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
124	123	rexbiia 3078	. . . . 5 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
125	124	abbii 2800	. . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
126	125	unieqi 4872	. . 3 ⊢ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(𝑓 ↾ 𝑦)))} = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
127	10, 11, 126	3eqtri 2760	. 2 ⊢ rec(𝐹, 𝑖) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
128	9, 127	vtoclg 3508	1 ⊢ (𝐼 ∈ 𝑉 → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {cab 2711   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  ifcif 4476  ∪ cuni 4860   ↦ cmpt 5176  dom cdm 5621  ran crn 5622   ↾ cres 5623   “ cima 5624  Rel wrel 5626  Ord word 6312  Oncon0 6313  Lim wlim 6314  suc csuc 6315   Fn wfn 6483  ‘cfv 6488  recscrecs 8298  reccrdg 8336

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7676

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fo 6494  df-fv 6496  df-ov 7357  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337

This theorem is referenced by:  dfrdg3  35861