Theorem oemapval 9706

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	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
oemapval.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
oemapval.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)

Assertion

Ref	Expression
oemapval	⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))

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

Step	Hyp	Ref	Expression
1		oemapval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
2		oemapval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
3		fveq1 6896	. . . . . 6 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑧) = (𝐹‘𝑧))
4		fveq1 6896	. . . . . 6 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑧) = (𝐺‘𝑧))
5		eleq12 2819	. . . . . 6 ⊢ (((𝑥‘𝑧) = (𝐹‘𝑧) ∧ (𝑦‘𝑧) = (𝐺‘𝑧)) → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧)))
6	3, 4, 5	syl2an 595	. . . . 5 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧)))
7		fveq1 6896	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤))
8		fveq1 6896	. . . . . . . 8 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑤) = (𝐺‘𝑤))
9	7, 8	eqeqan12d 2742	. . . . . . 7 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝐹‘𝑤) = (𝐺‘𝑤)))
10	9	imbi2d 340	. . . . . 6 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
11	10	ralbidv 3174	. . . . 5 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
12	6, 11	anbi12d 631	. . . 4 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
13	12	rexbidv 3175	. . 3 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
14		oemapval.t	. . 3 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
15	13, 14	brabga 5536	. 2 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
16	1, 2, 15	syl2anc 583	1 ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  ∃wrex 3067   class class class wbr 5148  {copab 5210  dom cdm 5678  Oncon0 6369  ‘cfv 6548  (class class class)co 7420   CNF ccnf 9684

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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6500  df-fv 6556

This theorem is referenced by:  oemapvali  9707