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Theorem elhoma 17280
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 17279 . . 3 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
87breqd 5068 . 2 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹))
9 brxp 5594 . . 3 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
10 opex 5347 . . . . 5 𝑋, 𝑌⟩ ∈ V
1110elsn2 4594 . . . 4 (𝑍 ∈ {⟨𝑋, 𝑌⟩} ↔ 𝑍 = ⟨𝑋, 𝑌⟩)
1211anbi1i 623 . . 3 ((𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
139, 12bitri 276 . 2 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
148, 13syl6bb 288 1 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {csn 4557  cop 4563   class class class wbr 5057   × cxp 5546  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  Catccat 16923  Homachoma 17271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-homa 17274
This theorem is referenced by:  elhomai  17281  homa1  17285  homahom2  17286
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