Theorem oemapval 9754

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 6921	. . . . . 6 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑧) = (𝐹‘𝑧))
4		fveq1 6921	. . . . . 6 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑧) = (𝐺‘𝑧))
5		eleq12 2834	. . . . . 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 6921	. . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤))
8		fveq1 6921	. . . . . . . 8 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑤) = (𝐺‘𝑤))
9	7, 8	eqeqan12d 2754	. . . . . . 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 3184	. . . . 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 3185	. . 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 5553	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   class class class wbr 5166  {copab 5228  dom cdm 5700  Oncon0 6397  ‘cfv 6575  (class class class)co 7450   CNF ccnf 9732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6527  df-fv 6583

This theorem is referenced by:  oemapvali  9755