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Theorem oemapval 8877
 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 6445 . . . . . 6 (𝑥 = 𝐹 → (𝑥𝑧) = (𝐹𝑧))
4 fveq1 6445 . . . . . 6 (𝑦 = 𝐺 → (𝑦𝑧) = (𝐺𝑧))
5 eleq12 2849 . . . . . 6 (((𝑥𝑧) = (𝐹𝑧) ∧ (𝑦𝑧) = (𝐺𝑧)) → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝐹𝑧) ∈ (𝐺𝑧)))
63, 4, 5syl2an 589 . . . . 5 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝐹𝑧) ∈ (𝐺𝑧)))
7 fveq1 6445 . . . . . . . 8 (𝑥 = 𝐹 → (𝑥𝑤) = (𝐹𝑤))
8 fveq1 6445 . . . . . . . 8 (𝑦 = 𝐺 → (𝑦𝑤) = (𝐺𝑤))
97, 8eqeqan12d 2794 . . . . . . 7 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝐹𝑤) = (𝐺𝑤)))
109imbi2d 332 . . . . . 6 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤))))
1110ralbidv 3168 . . . . 5 ((𝑥 = 𝐹𝑦 = 𝐺) → (∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤))))
126, 11anbi12d 624 . . . 4 ((𝑥 = 𝐹𝑦 = 𝐺) → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
1312rexbidv 3237 . . 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 5226 . 2 ((𝐹𝑆𝐺𝑆) → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
161, 2, 15syl2anc 579 1 (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091   class class class wbr 4886  {copab 4948  dom cdm 5355  Oncon0 5976  ‘cfv 6135  (class class class)co 6922   CNF ccnf 8855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-iota 6099  df-fv 6143 This theorem is referenced by:  oemapvali  8878
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