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Theorem oemapval 9726
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 6900 . . . . . 6 (𝑥 = 𝐹 → (𝑥𝑧) = (𝐹𝑧))
4 fveq1 6900 . . . . . 6 (𝑦 = 𝐺 → (𝑦𝑧) = (𝐺𝑧))
5 eleq12 2816 . . . . . 6 (((𝑥𝑧) = (𝐹𝑧) ∧ (𝑦𝑧) = (𝐺𝑧)) → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝐹𝑧) ∈ (𝐺𝑧)))
63, 4, 5syl2an 594 . . . . 5 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑥𝑧) ∈ (𝑦𝑧) ↔ (𝐹𝑧) ∈ (𝐺𝑧)))
7 fveq1 6900 . . . . . . . 8 (𝑥 = 𝐹 → (𝑥𝑤) = (𝐹𝑤))
8 fveq1 6900 . . . . . . . 8 (𝑦 = 𝐺 → (𝑦𝑤) = (𝐺𝑤))
97, 8eqeqan12d 2740 . . . . . . 7 ((𝑥 = 𝐹𝑦 = 𝐺) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝐹𝑤) = (𝐺𝑤)))
109imbi2d 339 . . . . . 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 630 . . . 4 ((𝑥 = 𝐹𝑦 = 𝐺) → (((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
1312rexbidv 3169 . . 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 5540 . 2 ((𝐹𝑆𝐺𝑆) → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
161, 2, 15syl2anc 582 1 (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧𝐵 ((𝐹𝑧) ∈ (𝐺𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝐹𝑤) = (𝐺𝑤)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  wrex 3060   class class class wbr 5153  {copab 5215  dom cdm 5682  Oncon0 6376  cfv 6554  (class class class)co 7424   CNF ccnf 9704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-iota 6506  df-fv 6562
This theorem is referenced by:  oemapvali  9727
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