Theorem dfrdg2 33490

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 8148	. . 3 ⊢ (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2		ifeq1 4443	. . . . . . . . 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 2748	. . . . . . . 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 3118	. . . . . . 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 632	. . . . . 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 3216	. . . . 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 2807	. . . 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 4833	. . 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 2753	. 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 8146	. . 3 ⊢ rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
11		dfrecs3 8109	. . 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 3412	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
13	12	resex 5899	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ 𝑦) ∈ V
14		eqeq1 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔 = ∅ ↔ (𝑓 ↾ 𝑦) = ∅))
15		relres 5880	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑓 ↾ 𝑦)
16		reldm0 5797	. . . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔 = ∅ ↔ dom (𝑓 ↾ 𝑦) = ∅))
19		dmeq 5772	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → dom 𝑔 = dom (𝑓 ↾ 𝑦))
20		limeq 6225	. . . . . . . . . . . . . . . 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 5805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = ran (𝑓 ↾ 𝑦))
23		df-ima 5564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
24	22, 23	eqtr4di 2796	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = (𝑓 “ 𝑦))
25	24	unieqd 4833	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ ran 𝑔 = ∪ (𝑓 “ 𝑦))
26		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → 𝑔 = (𝑓 ↾ 𝑦))
27	19	unieqd 4833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ dom 𝑔 = ∪ dom (𝑓 ↾ 𝑦))
28	26, 27	fveq12d 6724	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔‘∪ dom 𝑔) = ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))
29	28	fveq2d 6721	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))
30	21, 25, 29	ifbieq12d 4467	. . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 3412	. . . . . . . . . . . . . 14 ⊢ 𝑖 ∈ V
34		imaexg 7693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑦) ∈ V)
35	12, 34	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) ∈ V
36	35	uniex 7529	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝑓 “ 𝑦) ∈ V
37		fvex 6730	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) ∈ V
38	36, 37	ifex 4489	. . . . . . . . . . . . . 14 ⊢ if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))) ∈ V
39	33, 38	ifex 4489	. . . . . . . . . . . . 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 6818	. . . . . . . . . . . 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 5873	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ↾ 𝑦) = (𝑦 ∩ dom 𝑓)
43		onelss 6255	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
44	43	imp 410	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
45	44	3adant2 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
46		fndm 6481	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47	46	3ad2ant2 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom 𝑓 = 𝑥)
48	45, 47	sseqtrrd 3942	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ dom 𝑓)
49		df-ss 3883	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ dom 𝑓 ↔ (𝑦 ∩ dom 𝑓) = 𝑦)
50	48, 49	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∩ dom 𝑓) = 𝑦)
51	42, 50	syl5eq 2790	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom (𝑓 ↾ 𝑦) = 𝑦)
52		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (dom (𝑓 ↾ 𝑦) = ∅ ↔ 𝑦 = ∅))
53		limeq 6225	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (Lim dom (𝑓 ↾ 𝑦) ↔ Lim 𝑦))
54		unieq 4830	. . . . . . . . . . . . . . . . 17 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ∪ dom (𝑓 ↾ 𝑦) = ∪ 𝑦)
55	54	fveq2d 6721	. . . . . . . . . . . . . . . 16 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)) = ((𝑓 ↾ 𝑦)‘∪ 𝑦))
56	55	fveq2d 6721	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))
57	53, 56	ifbieq2d 4465	. . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . 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 6238	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
60		eloni 6223	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → Ord 𝑦)
61	59, 60	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
62	61	3adant2 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
63		ordzsl 7624	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64		iftrue 4445	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = 𝑖)
65		iftrue 4445	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = 𝑖)
66	64, 65	eqtr4d 2780	. . . . . . . . . . . . . . . 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 3412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
68	67	sucid 6292	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ suc 𝑧
69		fvres 6736	. . . . . . . . . . . . . . . . . . . . . 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 6223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ On → Ord 𝑧)
72		ordunisuc 7611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑧 → ∪ suc 𝑧 = 𝑧)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ On → ∪ suc 𝑧 = 𝑧)
74	73	fveq2d 6721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
75	73	fveq2d 6721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → (𝑓‘∪ suc 𝑧) = (𝑓‘𝑧))
76	70, 74, 75	3eqtr4a 2804	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = (𝑓‘∪ suc 𝑧))
77	76	fveq2d 6721	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
78		nsuceq0 6293	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑧 ≠ ∅
79	78	neii 2942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ suc 𝑧 = ∅
80	79	iffalsei 4449	. . . . . . . . . . . . . . . . . . . 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 7621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
82		iffalse 4448	. . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . 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 4449	. . . . . . . . . . . . . . . . . . . 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 4448	. . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
89	77, 84, 88	3eqtr4g 2803	. . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91		limeq 6225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92		reseq2 5846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → (𝑓 ↾ 𝑦) = (𝑓 ↾ suc 𝑧))
93		unieq 4830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
94	92, 93	fveq12d 6724	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → ((𝑓 ↾ 𝑦)‘∪ 𝑦) = ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))
95	94	fveq2d 6721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))
96	91, 95	ifbieq2d 4465	. . . . . . . . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . . . . . . . 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 6721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
99	98	fveq2d 6721	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
100	91, 99	ifbieq2d 4465	. . . . . . . . . . . . . . . . . . . 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 4465	. . . . . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . . . 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 250	. . . . . . . . . . . . . . . . 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 3199	. . . . . . . . . . . . . . . 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 4445	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
106		df-lim 6218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim 𝑦 ↔ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦))
107	106	simp2bi 1148	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝑦 → 𝑦 ≠ ∅)
108	107	neneqd 2945	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑦 → ¬ 𝑦 = ∅)
109	108	iffalsed 4450	. . . . . . . . . . . . . . . . . 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 4445	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
111	105, 109, 110	3eqtr4d 2787	. . . . . . . . . . . . . . . . 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 4450	. . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . 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 1429	. . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . 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	syl5eq 2790	. . . . . . . . . 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 2748	. . . . . . . . 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 1120	. . . . . . . 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 3117	. . . . . . 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 582	. . . . . 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 3169	. . . . 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 2808	. . . 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 4832	. . 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 2769	. 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 3481	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 209   ∧ wa 399   ∨ w3o 1088   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  ifcif 4439  ∪ cuni 4819   ↦ cmpt 5135  dom cdm 5551  ran crn 5552   ↾ cres 5553   “ cima 5554  Rel wrel 5556  Ord word 6212  Oncon0 6213  Lim wlim 6214  suc csuc 6215   Fn wfn 6375  ‘cfv 6380  recscrecs 8107  reccrdg 8145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-wrecs 8047  df-recs 8108  df-rdg 8146

This theorem is referenced by:  dfrdg3  33491