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Theorem erovlem 6629
Description: Lemma for eroprf 6630. (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 109 . . . . . . . 8 (((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆))
21reximi 2574 . . . . . . 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 2574 . . . . . 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 2244 . . . . . . . . 9 (ð‘Ĩ ∈ ð― ↔ ð‘Ĩ ∈ (ðī / 𝑅))
6 vex 2742 . . . . . . . . . 10 ð‘Ĩ ∈ V
76elqs 6588 . . . . . . . . 9 (ð‘Ĩ ∈ (ðī / 𝑅) ↔ ∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅)
85, 7bitri 184 . . . . . . . 8 (ð‘Ĩ ∈ ð― ↔ ∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅)
9 eropr.2 . . . . . . . . . 10 ðū = (ðĩ / 𝑆)
109eleq2i 2244 . . . . . . . . 9 (ð‘Ķ ∈ ðū ↔ ð‘Ķ ∈ (ðĩ / 𝑆))
11 vex 2742 . . . . . . . . . 10 ð‘Ķ ∈ V
1211elqs 6588 . . . . . . . . 9 (ð‘Ķ ∈ (ðĩ / 𝑆) ↔ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆)
1310, 12bitri 184 . . . . . . . 8 (ð‘Ķ ∈ ðū ↔ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆)
148, 13anbi12i 460 . . . . . . 7 ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ↔ (∃𝑝 ∈ ðī ð‘Ĩ = [𝑝]𝑅 ∧ ∃𝑞 ∈ ðĩ ð‘Ķ = [𝑞]𝑆))
15 reeanv 2647 . . . . . . 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 187 . . . . . 6 ((ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū) ↔ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ (ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆))
173, 16sylibr 134 . . . . 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 392 . . . 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 6628 . . . . . . 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 5194 . . . . . . 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 14 . . . . . 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 2179 . . . . . 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 196 . . . . 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 452 . . . 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 192 . . 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 5931 . 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 5882 . . 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 1528 . . . 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 1528 . . . . 5 â„ē𝑧(ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)
40 nfiota1 5182 . . . . . 6 â„ē𝑧(â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
4140nfeq2 2331 . . . . 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 1565 . . . 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 2184 . . . . 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 464 . . . 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 5953 . . 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 2201 . 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 2235 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 set class
Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  âˆƒ!weu 2026   ∈ wcel 2148  âˆƒwrex 2456   ⊆ wss 3131   class class class wbr 4005   × cxp 4626  â„Đcio 5178  âŸķwf 5214  (class class class)co 5877  {coprab 5878   ∈ cmpo 5879   Er wer 6534  [cec 6535   / cqs 6536
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-er 6537  df-ec 6539  df-qs 6543
This theorem is referenced by:  eroprf  6630
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