MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elhoma Structured version   Visualization version   GIF version

Theorem elhoma 16505
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 16504 . . 3 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
87breqd 4594 . 2 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹))
9 brxp 5071 . . 3 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
10 opex 4859 . . . . 5 𝑋, 𝑌⟩ ∈ V
1110elsn2 4158 . . . 4 (𝑍 ∈ {⟨𝑋, 𝑌⟩} ↔ 𝑍 = ⟨𝑋, 𝑌⟩)
1211anbi1i 727 . . 3 ((𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
139, 12bitri 263 . 2 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
148, 13syl6bb 275 1 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {csn 4125  cop 4131   class class class wbr 4583   × cxp 5036  cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  Homachoma 16496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-homa 16499
This theorem is referenced by:  elhomai  16506  homa1  16510  homahom2  16511
  Copyright terms: Public domain W3C validator