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Theorem erovlem 8803
Description: Lemma for erov 8804 and eroprf 8805. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
eropr.1 ð― = (ðī / 𝑅)
eropr.2 ðū = (ðĩ / 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 → ðī ⊆ 𝑈)
eropr.8 (𝜑 → ðĩ ⊆ 𝑉)
eropr.9 (𝜑 → ðķ ⊆ 𝑊)
eropr.10 (𝜑 → + :(ðī × ðĩ)âŸķðķ)
eropr.11 ((𝜑 ∧ ((𝑟 ∈ ðī ∧ 𝑠 ∈ ðī) ∧ (ð‘Ą ∈ ðĩ ∧ ð‘Ē ∈ ðĩ))) → ((𝑟𝑅𝑠 ∧ ð‘Ąð‘†ð‘Ē) → (𝑟 + ð‘Ą)𝑇(𝑠 + ð‘Ē)))
eropr.12 âĻĢ = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
Assertion
Ref Expression
erovlem (𝜑 → âĻĢ = (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧,ðī   ðĩ,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   ð―,𝑝,𝑞,ð‘Ĩ,ð‘Ķ,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   ðū,𝑝,𝑞,ð‘Ĩ,ð‘Ķ,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   + ,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧
Allowed substitution hints:   ðķ(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   âĻĢ (ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑈(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   ð―(ð‘Ē,ð‘Ą,𝑠,𝑟)   ðū(ð‘Ē,ð‘Ą,𝑠,𝑟)   𝑉(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑊(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑍(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)
Proof of Theorem erovlem
Dummy variable ð‘Ī is distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . . . 8 (((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆))
21reximi 3084 . . . . . . 7 (∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → ∃𝑞 ∈ ðĩ (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆))
32reximi 3084 . . . . . 6 (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆))
4 eropr.1 . . . . . . . . . 10 ð― = (ðī / 𝑅)
54eleq2i 2825 . . . . . . . . 9 (ð‘Ĩ ∈ ð― ↔ ð‘Ĩ ∈ (ðī / 𝑅))
6 vex 3478 . . . . . . . . . 10 ð‘Ĩ ∈ V
76elqs 8759 . . . . . . . . 9 (ð‘Ĩ ∈ (ðī / 𝑅) ↔ ∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅)
85, 7bitri 274 . . . . . . . 8 (ð‘Ĩ ∈ ð― ↔ ∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅)
9 eropr.2 . . . . . . . . . 10 ðū = (ðĩ / 𝑆)
109eleq2i 2825 . . . . . . . . 9 (ð‘Ķ ∈ ðū ↔ ð‘Ķ ∈ (ðĩ / 𝑆))
11 vex 3478 . . . . . . . . . 10 ð‘Ķ ∈ V
1211elqs 8759 . . . . . . . . 9 (ð‘Ķ ∈ (ðĩ / 𝑆) ↔ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆)
1310, 12bitri 274 . . . . . . . 8 (ð‘Ķ ∈ ðū ↔ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆)
148, 13anbi12i 627 . . . . . . 7 ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ↔ (∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅 ∧ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆))
15 reeanv 3226 . . . . . . 7 (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ↔ (∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅 ∧ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆))
1614, 15bitr4i 277 . . . . . 6 ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ↔ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆))
173, 16sylibr 233 . . . . 5 (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū))
1817pm4.71ri 561 . . . 4 (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [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 → ðī ⊆ 𝑈)
24 eropr.8 . . . . . . . 8 (𝜑 → ðĩ ⊆ 𝑉)
25 eropr.9 . . . . . . . 8 (𝜑 → ðķ ⊆ 𝑊)
26 eropr.10 . . . . . . . 8 (𝜑 → + :(ðī × ðĩ)âŸķðķ)
27 eropr.11 . . . . . . . 8 ((𝜑 ∧ ((𝑟 ∈ ðī ∧ 𝑠 ∈ ðī) ∧ (ð‘Ą ∈ ðĩ ∧ ð‘Ē ∈ ðĩ))) → ((𝑟𝑅𝑠 ∧ ð‘Ąð‘†ð‘Ē) → (𝑟 + ð‘Ą)𝑇(𝑠 + ð‘Ē)))
284, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27eroveu 8802 . . . . . . 7 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → ∃!𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
29 iota1 6517 . . . . . . 7 (∃!𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = 𝑧))
3028, 29syl 17 . . . . . 6 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = 𝑧))
31 eqcom 2739 . . . . . 6 ((â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = 𝑧 ↔ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
3230, 31bitrdi 286 . . . . 5 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
3332pm5.32da 579 . . . 4 (𝜑 → (((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ↔ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))))
3418, 33bitrid 282 . . 3 (𝜑 → (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))))
3534oprabbidv 7471 . 2 (𝜑 → {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))})
36 eropr.12 . 2 âĻĢ = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
37 df-mpo 7410 . . 3 (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, ð‘ĪâŸĐ âˆĢ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ ð‘Ī = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))}
38 nfv 1917 . . . 4 â„ēð‘Ī((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
39 nfv 1917 . . . . 5 â„ē𝑧(ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)
40 nfiota1 6494 . . . . . 6 â„ē𝑧(â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
4140nfeq2 2920 . . . . 5 â„ē𝑧 ð‘Ī = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
4239, 41nfan 1902 . . . 4 â„ē𝑧((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ ð‘Ī = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
43 eqeq1 2736 . . . . 5 (𝑧 = ð‘Ī → (𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ↔ ð‘Ī = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
4443anbi2d 629 . . . 4 (𝑧 = ð‘Ī → (((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) ↔ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ ð‘Ī = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))))
4538, 42, 44cbvoprab3 7496 . . 3 {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))} = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, ð‘ĪâŸĐ âˆĢ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ ð‘Ī = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))}
4637, 45eqtr4i 2763 . 2 (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ∧ 𝑧 = (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))}
4735, 36, 463eqtr4g 2797 1 (𝜑 → âĻĢ = (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  âˆƒ!weu 2562  âˆƒwrex 3070   ⊆ wss 3947   class class class wbr 5147   × cxp 5673  â„Đcio 6490  âŸķwf 6536  (class class class)co 7405  {coprab 7406   ∈ cmpo 7407   Er wer 8696  [cec 8697   / cqs 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8699  df-ec 8701  df-qs 8705
This theorem is referenced by:  erov  8804  eroprf  8805
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