Theorem elhoma 17286

Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homarcl.h	⊢ 𝐻 = (Homa‘𝐶)
homafval.b	⊢ 𝐵 = (Base‘𝐶)
homafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
homaval.j	⊢ 𝐽 = (Hom ‘𝐶)
homaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
homaval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
elhoma	⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))

Proof of Theorem elhoma

Step	Hyp	Ref	Expression
1		homarcl.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
2		homafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		homafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		homaval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐶)
5		homaval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		homaval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	1, 2, 3, 4, 5, 6	homaval 17285	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
8	7	breqd 5069	. 2 ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ 𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹))
9		brxp 5595	. . 3 ⊢ (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
10		opex 5348	. . . . 5 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
11	10	elsn2 4597	. . . 4 ⊢ (𝑍 ∈ {⟨𝑋, 𝑌⟩} ↔ 𝑍 = ⟨𝑋, 𝑌⟩)
12	11	anbi1i 625	. . 3 ⊢ ((𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
13	9, 12	bitri 277	. 2 ⊢ (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
14	8, 13	syl6bb 289	1 ⊢ (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {csn 4560  ⟨cop 4566   class class class wbr 5058   × cxp 5547  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  Hom chom 16570  Catccat 16929  Hom_achoma 17277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-homa 17280

This theorem is referenced by:  elhomai  17287  homa1  17291  homahom2  17292