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

Theorem elhoma 16451
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 16450 . . 3 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
87breqd 4588 . 2 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹))
9 brxp 5061 . . 3 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
10 opex 4853 . . . . 5 𝑋, 𝑌⟩ ∈ V
1110elsn2 4157 . . . 4 (𝑍 ∈ {⟨𝑋, 𝑌⟩} ↔ 𝑍 = ⟨𝑋, 𝑌⟩)
1211anbi1i 726 . . 3 ((𝑍 ∈ {⟨𝑋, 𝑌⟩} ∧ 𝐹 ∈ (𝑋𝐽𝑌)) ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
139, 12bitri 262 . 2 (𝑍({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌))𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
148, 13syl6bb 274 1 (𝜑 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {csn 4124  cop 4130   class class class wbr 4577   × cxp 5026  cfv 5790  (class class class)co 6527  Basecbs 15641  Hom chom 15725  Catccat 16094  Homachoma 16442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-homa 16445
This theorem is referenced by:  elhomai  16452  homa1  16456  homahom2  16457
  Copyright terms: Public domain W3C validator