Theorem erovlem 6891

Description: Lemma for eroprf 6892. (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 2647	. . . . . . 7 ⊢ (∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → ∃𝑞 ∈ 𝐵 (𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆))
3	2	reximi 2647	. . . . . 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 2305	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐽 ↔ 𝑥 ∈ (𝐴 / 𝑅))
6		vex 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
7	6	elqs 6850	. . . . . . . . 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 2305	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐾 ↔ 𝑦 ∈ (𝐵 / 𝑆))
11		vex 2824	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
12	11	elqs 6850	. . . . . . . . 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 464	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ↔ (∃𝑝 ∈ 𝐴 𝑥 = [𝑝]𝑅 ∧ ∃𝑞 ∈ 𝐵 𝑦 = [𝑞]𝑆))
15		reeanv 2721	. . . . . . 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 396	. . . 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 6890	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
29		iota1 5347	. . . . . . 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 2240	. . . . . 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 456	. . . 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 6132	. 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 6080	. . 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 1581	. . . 4 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
39		nfv 1581	. . . . 5 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)
40		nfiota1 5334	. . . . . 6 ⊢ Ⅎ𝑧(℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
41	40	nfeq2 2404	. . . . 5 ⊢ Ⅎ𝑧 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
42	39, 41	nfan 1618	. . . 4 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾) ∧ 𝑤 = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
43		eqeq1 2245	. . . . 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 468	. . . 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 6154	. . 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 2262	. 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 2296	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 1402  ∃!weu 2086   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220   class class class wbr 4125   × cxp 4767  ℩cio 5330  ⟶wf 5368  (class class class)co 6075  {coprab 6076   ∈ cmpo 6077   Er wer 6794  [cec 6795   / cqs 6796

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-er 6797  df-ec 6799  df-qs 6803

This theorem is referenced by:  eroprf  6892