Theorem dfrdg2 36266

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 8400	. . 3 ⊢ (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2		ifeq1 4492	. . . . . . . . 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 2774	. . . . . . . 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 3188	. . . . . . 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 641	. . . . . 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 3189	. . . . 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 2829	. . . 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 4886	. . 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 2779	. 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 8398	. . 3 ⊢ rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
11		dfrecs3 8360	. . 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 3459	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
13	12	resex 6030	. . . . . . . . . . . 12 ⊢ (𝑓 ↾ 𝑦) ∈ V
14		eqeq1 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔 = ∅ ↔ (𝑓 ↾ 𝑦) = ∅))
15		relres 6006	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝑓 ↾ 𝑦)
16		reldm0 5920	. . . . . . . . . . . . . . . 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 5895	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → dom 𝑔 = dom (𝑓 ↾ 𝑦))
20		limeq 6374	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝑔 = dom (𝑓 ↾ 𝑦) → (Lim dom 𝑔 ↔ Lim dom (𝑓 ↾ 𝑦)))
21	19, 20	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (Lim dom 𝑔 ↔ Lim dom (𝑓 ↾ 𝑦)))
22		rneq 5928	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = ran (𝑓 ↾ 𝑦))
23		df-ima 5676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦)
24	22, 23	eqtr4di 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ran 𝑔 = (𝑓 “ 𝑦))
25	24	unieqd 4886	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ ran 𝑔 = ∪ (𝑓 “ 𝑦))
26		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → 𝑔 = (𝑓 ↾ 𝑦))
27	19	unieqd 4886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → ∪ dom 𝑔 = ∪ dom (𝑓 ↾ 𝑦))
28	26, 27	fveq12d 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝑔‘∪ dom 𝑔) = ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))
29	28	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ 𝑦) → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))))
30	21, 25, 29	ifbieq12d 4517	. . . . . . . . . . . . . 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 4515	. . . . . . . . . . . . 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 2763	. . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . 14 ⊢ 𝑖 ∈ V
34		imaexg 7911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ V → (𝑓 “ 𝑦) ∈ V)
35	12, 34	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 “ 𝑦) ∈ V
36	35	uniex 7741	. . . . . . . . . . . . . . 15 ⊢ ∪ (𝑓 “ 𝑦) ∈ V
37		fvex 6896	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) ∈ V
38	36, 37	ifex 4539	. . . . . . . . . . . . . 14 ⊢ if(Lim dom (𝑓 ↾ 𝑦), ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)))) ∈ V
39	33, 38	ifex 4539	. . . . . . . . . . . . 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 6991	. . . . . . . . . . . 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 6013	. . . . . . . . . . . . 13 ⊢ dom (𝑓 ↾ 𝑦) = (𝑦 ∩ dom 𝑓)
43		onelss 6405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
44	43	imp 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
45	44	3adant2 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ 𝑥)
46		fndm 6640	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47	46	3ad2ant2 1152	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom 𝑓 = 𝑥)
48	45, 47	sseqtrrd 3975	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → 𝑦 ⊆ dom 𝑓)
49		dfss2 3924	. . . . . . . . . . . . . 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	eqtrid 2810	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → dom (𝑓 ↾ 𝑦) = 𝑦)
52		eqeq1 2767	. . . . . . . . . . . . . 14 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (dom (𝑓 ↾ 𝑦) = ∅ ↔ 𝑦 = ∅))
53		limeq 6374	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (Lim dom (𝑓 ↾ 𝑦) ↔ Lim 𝑦))
54		unieq 4884	. . . . . . . . . . . . . . . . 17 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ∪ dom (𝑓 ↾ 𝑦) = ∪ 𝑦)
55	54	fveq2d 6887	. . . . . . . . . . . . . . . 16 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → ((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦)) = ((𝑓 ↾ 𝑦)‘∪ 𝑦))
56	55	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (dom (𝑓 ↾ 𝑦) = 𝑦 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ dom (𝑓 ↾ 𝑦))) = (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))
57	53, 56	ifbieq2d 4515	. . . . . . . . . . . . . 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 4515	. . . . . . . . . . . . 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 6387	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
60		eloni 6372	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ On → Ord 𝑦)
61	59, 60	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
62	61	3adant2 1149	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
63		ordzsl 7842	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64		iftrue 4494	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)))) = 𝑖)
65		iftrue 4494	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = 𝑖)
66	64, 65	eqtr4d 2801	. . . . . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
68	67	sucid 6447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑧 ∈ suc 𝑧
69		fvres 6902	. . . . . . . . . . . . . . . . . . . . . 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 6372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ On → Ord 𝑧)
72		ordunisuc 7829	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑧 → ∪ suc 𝑧 = 𝑧)
73	71, 72	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ On → ∪ suc 𝑧 = 𝑧)
74	73	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
75	73	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ On → (𝑓‘∪ suc 𝑧) = (𝑓‘𝑧))
76	70, 74, 75	3eqtr4a 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧) = (𝑓‘∪ suc 𝑧))
77	76	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
78		nsuceq0 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑧 ≠ ∅
79	78	neii 2960	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ¬ suc 𝑧 = ∅
80	79	iffalsei 4498	. . . . . . . . . . . . . . . . . . . 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 7839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ V → ¬ Lim suc 𝑧)
82		iffalse 4497	. . . . . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . . . . . 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 4497	. . . . . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
89	77, 84, 88	3eqtr4g 2823	. . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91		limeq 6374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92		reseq2 5975	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → (𝑓 ↾ 𝑦) = (𝑓 ↾ suc 𝑧))
93		unieq 4884	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
94	92, 93	fveq12d 6890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → ((𝑓 ↾ 𝑦)‘∪ 𝑦) = ((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧))
95	94	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘∪ suc 𝑧)))
96	91, 95	ifbieq2d 4515	. . . . . . . . . . . . . . . . . . . 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 4515	. . . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
99	98	fveq2d 6887	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
100	91, 99	ifbieq2d 4515	. . . . . . . . . . . . . . . . . . . 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 4515	. . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘((𝑓 ↾ 𝑦)‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
106		df-lim 6367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim 𝑦 ↔ (Ord 𝑦 ∧ 𝑦 ≠ ∅ ∧ 𝑦 = ∪ 𝑦))
107	106	simp2bi 1164	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝑦 → 𝑦 ≠ ∅)
108	107	neneqd 2963	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝑦 → ¬ 𝑦 = ∅)
109	108	iffalsed 4499	. . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
111	105, 109, 110	3eqtr4d 2808	. . . . . . . . . . . . . . . . 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 4499	. . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 2820	. . . . . . . . . . . 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 699	. . . . . . . . . . 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 2810	. . . . . . . . . 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 2774	. . . . . . . . 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 1136	. . . . . . . 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 3186	. . . . . . 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 589	. . . . . 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 3110	. . . . 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 2830	. . . 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 4885	. . 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 2790	. 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 3523	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 400   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  ifcif 4488  ∪ cuni 4873   ↦ cmpt 5193  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666  Rel wrel 5668  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  recscrecs 8358  reccrdg 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7415  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398

This theorem is referenced by:  dfrdg3  36267