Theorem dfrdg2 35759

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 8468	. . 3 ⊢ (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2		ifeq1 4552	. . . . . . . . 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 2751	. . . . . . . 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 3184	. . . . . . 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 629	. . . . . 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 3185	. . . . 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 2811	. . . 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 4944	. . 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 2756	. 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 8466	. . 3 ⊢ rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
11		dfrecs3 8428	. . 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 3492	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
13	12	resex 6058	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ 𝑦) ∈ V
14		eqeq1 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔 = ∅ ↔ (𝑓 ↾ 𝑦) = ∅))
15		relres 6035	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑓 ↾ 𝑦)
16		reldm0 5952	. . . . . . . . . . . . . . . 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 5928	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → dom 𝑔 = dom (𝑓 ↾ 𝑦))
20		limeq 6407	. . . . . . . . . . . . . . . 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 5961	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = ran (𝑓 ↾ 𝑦))
23		df-ima 5713	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
24	22, 23	eqtr4di 2798	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = (𝑓 “ 𝑦))
25	24	unieqd 4944	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ ran 𝑔 = ∪ (𝑓 “ 𝑦))
26		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → 𝑔 = (𝑓 ↾ 𝑦))
27	19	unieqd 4944	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ dom 𝑔 = ∪ dom (𝑓 ↾ 𝑦))
28	26, 27	fveq12d 6927	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔‘∪ dom 𝑔) = ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))
29	28	fveq2d 6924	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))
30	21, 25, 29	ifbieq12d 4576	. . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 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 3492	. . . . . . . . . . . . . 14 ⊢ 𝑖 ∈ V
34		imaexg 7953	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑦) ∈ V)
35	12, 34	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) ∈ V
36	35	uniex 7776	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝑓 “ 𝑦) ∈ V
37		fvex 6933	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) ∈ V
38	36, 37	ifex 4598	. . . . . . . . . . . . . 14 ⊢ if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))) ∈ V
39	33, 38	ifex 4598	. . . . . . . . . . . . 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 7029	. . . . . . . . . . . 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 6041	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ↾ 𝑦) = (𝑦 ∩ dom 𝑓)
43		onelss 6437	. . . . . . . . . . . . . . . . 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 6682	. . . . . . . . . . . . . . . 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 4050	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ dom 𝑓)
49		dfss2 3994	. . . . . . . . . . . . . 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 2792	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom (𝑓 ↾ 𝑦) = 𝑦)
52		eqeq1 2744	. . . . . . . . . . . . . 14 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (dom (𝑓 ↾ 𝑦) = ∅ ↔ 𝑦 = ∅))
53		limeq 6407	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (Lim dom (𝑓 ↾ 𝑦) ↔ Lim 𝑦))
54		unieq 4942	. . . . . . . . . . . . . . . . 17 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ∪ dom (𝑓 ↾ 𝑦) = ∪ 𝑦)
55	54	fveq2d 6924	. . . . . . . . . . . . . . . 16 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)) = ((𝑓 ↾ 𝑦)‘∪ 𝑦))
56	55	fveq2d 6924	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))
57	53, 56	ifbieq2d 4574	. . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . 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 6420	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
60		eloni 6405	. . . . . . . . . . . . . . . 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 7882	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64		iftrue 4554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = 𝑖)
65		iftrue 4554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = 𝑖)
66	64, 65	eqtr4d 2783	. . . . . . . . . . . . . . . 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 3492	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
68	67	sucid 6477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ suc 𝑧
69		fvres 6939	. . . . . . . . . . . . . . . . . . . . . 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 6405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ On → Ord 𝑧)
72		ordunisuc 7868	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑧 → ∪ suc 𝑧 = 𝑧)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ On → ∪ suc 𝑧 = 𝑧)
74	73	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
75	73	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → (𝑓‘∪ suc 𝑧) = (𝑓‘𝑧))
76	70, 74, 75	3eqtr4a 2806	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = (𝑓‘∪ suc 𝑧))
77	76	fveq2d 6924	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
78		nsuceq0 6478	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑧 ≠ ∅
79	78	neii 2948	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ suc 𝑧 = ∅
80	79	iffalsei 4558	. . . . . . . . . . . . . . . . . . . 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 7879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
82		iffalse 4557	. . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . 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 4558	. . . . . . . . . . . . . . . . . . . 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 4557	. . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . 19 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
89	77, 84, 88	3eqtr4g 2805	. . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91		limeq 6407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92		reseq2 6004	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → (𝑓 ↾ 𝑦) = (𝑓 ↾ suc 𝑧))
93		unieq 4942	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
94	92, 93	fveq12d 6927	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → ((𝑓 ↾ 𝑦)‘∪ 𝑦) = ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))
95	94	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))
96	91, 95	ifbieq2d 4574	. . . . . . . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
99	98	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
100	91, 99	ifbieq2d 4574	. . . . . . . . . . . . . . . . . . . 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 4574	. . . . . . . . . . . . . . . . . . 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 2756	. . . . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . . 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 4554	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
106		df-lim 6400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim 𝑦 ↔ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦))
107	106	simp2bi 1146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝑦 → 𝑦 ≠ ∅)
108	107	neneqd 2951	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑦 → ¬ 𝑦 = ∅)
109	108	iffalsed 4559	. . . . . . . . . . . . . . . . . 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 4554	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
111	105, 109, 110	3eqtr4d 2790	. . . . . . . . . . . . . . . . 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 4559	. . . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . . 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 1428	. . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . 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 686	. . . . . . . . . . 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 2792	. . . . . . . . . 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 2751	. . . . . . . . 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 3182	. . . . . . 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 578	. . . . . 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 3098	. . . . 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 2812	. . . 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 4943	. . 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 2772	. 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 3566	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 1086   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ifcif 4548  ∪ cuni 4931   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Rel wrel 5705  Ord word 6394  Oncon0 6395  Lim wlim 6396  suc csuc 6397   Fn wfn 6568  ‘cfv 6573  recscrecs 8426  reccrdg 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466

This theorem is referenced by:  dfrdg3  35760