Theorem erovlem 6653

Description: Lemma for eroprf 6654. (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 109	. . . . . . . 8 ⊢ (((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
2	1	reximi 2587	. . . . . . 7 ⊢ (∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
3	2	reximi 2587	. . . . . 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 2256	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ (𝐴 / 𝑅))
6		vex 2755	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7	6	elqs 6612	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅)
8	5, 7	bitri 184	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐽 ↔ ∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅)
9		eropr.2	. . . . . . . . . 10 ⊢ 𝐾 = (𝐵 / 𝑆)
10	9	eleq2i 2256	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐾 ↔ 𝑦 ∈ (𝐵 / 𝑆))
11		vex 2755	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
12	11	elqs 6612	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 / 𝑆) ↔ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆)
13	10, 12	bitri 184	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐾 ↔ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆)
14	8, 13	anbi12i 460	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ↔ (∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆))
15		reeanv 2660	. . . . . . 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 187	. . . . . 6 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
17	3, 16	sylibr 134	. . . . 5 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾))
18	17	pm4.71ri 392	. . . 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 6652	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
29		iota1 5210	. . . . . . 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 14	. . . . . 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 2191	. . . . . 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 196	. . . . 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 452	. . . 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 192	. . 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 5950	. 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 5901	. . 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 1539	. . . 4 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
39		nfv 1539	. . . . 5 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)
40		nfiota1 5198	. . . . . 6 ⊢ Ⅎ𝑧(℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
41	40	nfeq2 2344	. . . . 5 ⊢ Ⅎ𝑧 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
42	39, 41	nfan 1576	. . . 4 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
43		eqeq1 2196	. . . . 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 464	. . . 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 5972	. . 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 2213	. 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 2247	1 ⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃!weu 2038   ∈ wcel 2160  ∃wrex 2469   ⊆ wss 3144   class class class wbr 4018   × cxp 4642  ℩cio 5194  ⟶wf 5231  (class class class)co 5896  {coprab 5897   ∈ cmpo 5898   Er wer 6556  [cec 6557   / cqs 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-er 6559  df-ec 6561  df-qs 6565

This theorem is referenced by:  eroprf  6654