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Theorem elhoma 16903
Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h 𝐻 = (Homa𝐶)
homafval.b 𝐵 = (Base‘𝐶)
homafval.c (𝜑𝐶 ∈ Cat)
homaval.j 𝐽 = (Hom ‘𝐶)
homaval.x (𝜑𝑋𝐵)
homaval.y (𝜑𝑌𝐵)
Assertion
Ref Expression
elhoma (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))

Proof of Theorem elhoma
StepHypRef Expression
1 homarcl.h . . . 4 𝐻 = (Homa𝐶)
2 homafval.b . . . 4 𝐵 = (Base‘𝐶)
3 homafval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 homaval.j . . . 4 𝐽 = (Hom ‘𝐶)
5 homaval.x . . . 4 (𝜑𝑋𝐵)
6 homaval.y . . . 4 (𝜑𝑌𝐵)
71, 2, 3, 4, 5, 6homaval 16902 . . 3 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
87breqd 4815 . 2 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹))
9 brxp 5304 . . 3 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
10 opex 5081 . . . . 5 𝑋, 𝑌⟩ ∈ V
1110elsn2 4356 . . . 4 (𝑍 ∈ {⟨𝑋, 𝑌⟩} ↔ 𝑍 = ⟨𝑋, 𝑌⟩)
1211anbi1i 733 . . 3 ((𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
139, 12bitri 264 . 2 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
148, 13syl6bb 276 1 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {csn 4321  cop 4327   class class class wbr 4804   × cxp 5264  cfv 6049  (class class class)co 6814  Basecbs 16079  Hom chom 16174  Catccat 16546  Homachoma 16894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-homa 16897
This theorem is referenced by:  elhomai  16904  homa1  16908  homahom2  16909
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