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Theorem oemapval 8540
Description: Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
oemapval.f (𝜑𝐹𝑆)
oemapval.g (𝜑𝐺𝑆)
Assertion
Ref Expression
oemapval (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐵   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐹,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑤,𝐺,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem oemapval
StepHypRef Expression
1 oemapval.f . 2 (𝜑𝐹𝑆)
2 oemapval.g . 2 (𝜑𝐺𝑆)
3 fveq1 6157 . . . . . 6 (𝑥 = 𝐹 → (𝑥𝑧) = (𝐹𝑧))
4 fveq1 6157 . . . . . 6 (𝑦 = 𝐺 → (𝑦𝑧) = (𝐺𝑧))
5 eleq12 2688 . . . . . 6 (((𝑥𝑧) = (𝐹𝑧) ∧ (𝑦𝑧) = (𝐺𝑧)) → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝐹𝑧) ∈ (𝐺𝑧)))
63, 4, 5syl2an 494 . . . . 5 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝐹𝑧) ∈ (𝐺𝑧)))
7 fveq1 6157 . . . . . . . 8 (𝑥 = 𝐹 → (𝑥𝑤) = (𝐹𝑤))
8 fveq1 6157 . . . . . . . 8 (𝑦 = 𝐺 → (𝑦𝑤) = (𝐺𝑤))
97, 8eqeqan12d 2637 . . . . . . 7 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝐹𝑤) = (𝐺𝑤)))
109imbi2d 330 . . . . . 6 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤))))
1110ralbidv 2982 . . . . 5 ((𝑥 = 𝐹𝑦 = 𝐺) → (∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤))))
126, 11anbi12d 746 . . . 4 ((𝑥 = 𝐹𝑦 = 𝐺) → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
1312rexbidv 3047 . . 3 ((𝑥 = 𝐹𝑦 = 𝐺) → (∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
14 oemapval.t . . 3 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
1513, 14brabga 4959 . 2 ((𝐹𝑆𝐺𝑆) → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
161, 2, 15syl2anc 692 1 (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909   class class class wbr 4623  {copab 4682  dom cdm 5084  Oncon0 5692  cfv 5857  (class class class)co 6615   CNF ccnf 8518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-iota 5820  df-fv 5865
This theorem is referenced by:  oemapvali  8541
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