Theorem erovlem 8817

Description: Lemma for erov 8818 and eroprf 8819. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
eropr.1	⊢ 𝐽 = (𝐴 / 𝑅)
eropr.2	⊢ 𝐾 = (𝐵 / 𝑆)
eropr.3	⊢ (𝜑 → 𝑇 ∈ 𝑍)
eropr.4	⊢ (𝜑 → 𝑅 Er 𝑈)
eropr.5	⊢ (𝜑 → 𝑆 Er 𝑉)
eropr.6	⊢ (𝜑 → 𝑇 Er 𝑊)
eropr.7	⊢ (𝜑 → 𝐴 ⊆ 𝑈)
eropr.8	⊢ (𝜑 → 𝐵 ⊆ 𝑉)
eropr.9	⊢ (𝜑 → 𝐶 ⊆ 𝑊)
eropr.10	⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)
eropr.11	⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
eropr.12	⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}

Assertion

Ref	Expression
erovlem	⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))

Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐾,𝑝,𝑞,𝑥,𝑦,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐶(𝑥, 𝑦, 𝑧, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝)   ⨣ (𝑥, 𝑦, 𝑧, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝)   𝑈(𝑥, 𝑦, 𝑧, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝)   𝐽(𝑢, 𝑡, 𝑠, 𝑟)   𝐾(𝑢, 𝑡, 𝑠, 𝑟)   𝑉(𝑥, 𝑦, 𝑧, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝)   𝑊(𝑥, 𝑦, 𝑧, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝)   𝑍(𝑥, 𝑦, 𝑧, 𝑢, 𝑡, 𝑠, 𝑟, 𝑞, 𝑝)

Proof of Theorem erovlem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . . . . 8 ⊢ (((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
2	1	reximi 3105	. . . . . . 7 ⊢ (∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
3	2	reximi 3105	. . . . . 6 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
4		eropr.1	. . . . . . . . . 10 ⊢ 𝐽 = (𝐴 / 𝑅)
5	4	eleq2i 2857	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ (𝐴 / 𝑅))
6		vex 3461	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7	6	elqs 8768	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅)
8	5, 7	bitri 278	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 ↔ ∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅)
9		eropr.2	. . . . . . . . . 10 ⊢ 𝐾 = (𝐵 / 𝑆)
10	9	eleq2i 2857	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐾 ↔ 𝑦 ∈ (𝐵 / 𝑆))
11		vex 3461	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
12	11	elqs 8768	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 / 𝑆) ↔ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆)
13	10, 12	bitri 278	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐾 ↔ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆)
14	8, 13	anbi12i 640	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ↔ (∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆))
15		reeanv 3239	. . . . . . 7 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ↔ (∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆))
16	14, 15	bitr4i 281	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
17	3, 16	sylibr 237	. . . . 5 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾))
18	17	pm4.71ri 570	. . . 4 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
19		eropr.3	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ 𝑍)
20		eropr.4	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Er 𝑈)
21		eropr.5	. . . . . . . 8 ⊢ (𝜑 → 𝑆 Er 𝑉)
22		eropr.6	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Er 𝑊)
23		eropr.7	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
24		eropr.8	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
25		eropr.9	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ 𝑊)
26		eropr.10	. . . . . . . 8 ⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶)
27		eropr.11	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
28	4, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27	eroveu 8816	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
29		iota1 6519	. . . . . . 7 ⊢ (∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = 𝑧))
30	28, 29	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = 𝑧))
31		eqcom 2772	. . . . . 6 ⊢ ((℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = 𝑧 ↔ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
32	30, 31	bitrdi 290	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
33	32	pm5.32da 590	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))))
34	18, 33	bitrid 286	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))))
35	34	oprabbidv 7485	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))})
36		eropr.12	. 2 ⊢ ⨣ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
37		df-mpo 7424	. . 3 ⊢ (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))}
38		nfv 1947	. . . 4 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
39		nfv 1947	. . . . 5 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)
40		nfiota1 6498	. . . . . 6 ⊢ Ⅎ𝑧(℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
41	40	nfeq2 2944	. . . . 5 ⊢ Ⅎ𝑧 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
42	39, 41	nfan 1932	. . . 4 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
43		eqeq1 2769	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ↔ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
44	43	anbi2d 642	. . . 4 ⊢ (𝑧 = 𝑤 → (((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) ↔ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))))
45	38, 42, 44	cbvoprab3 7510	. . 3 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))}
46	37, 45	eqtr4i 2791	. 2 ⊢ (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))}
47	35, 36, 46	3eqtr4g 2825	1 ⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  ∃wrex 3091   ⊆ wss 3906   class class class wbr 5111   × cxp 5661  ℩cio 6494  ⟶wf 6536  (class class class)co 7419  {coprab 7420   ∈ cmpo 7421   Er wer 8697  [cec 8698   / cqs 8699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  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-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-er 8700  df-ec 8702  df-qs 8706

This theorem is used by:  erov  8818  eroprf  8819