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Theorem homaval 16504
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
homaval (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))

Proof of Theorem homaval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 6552 . 2 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
2 homarcl.h . . . 4 𝐻 = (Homa𝐶)
3 homafval.b . . . 4 𝐵 = (Base‘𝐶)
4 homafval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 homaval.j . . . 4 𝐽 = (Hom ‘𝐶)
62, 3, 4, 5homafval 16502 . . 3 (𝜑𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽𝑧))))
7 simpr 476 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
87sneqd 4137 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
97fveq2d 6107 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10 df-ov 6552 . . . . 5 (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
119, 10syl6eqr 2662 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝑋𝐽𝑌))
128, 11xpeq12d 5064 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ({𝑧} × (𝐽𝑧)) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
13 homaval.x . . . 4 (𝜑𝑋𝐵)
14 homaval.y . . . 4 (𝜑𝑌𝐵)
15 opelxpi 5072 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
1613, 14, 15syl2anc 691 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
17 snex 4835 . . . . 5 {⟨𝑋, 𝑌⟩} ∈ V
18 ovex 6577 . . . . 5 (𝑋𝐽𝑌) ∈ V
1917, 18xpex 6860 . . . 4 ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
2019a1i 11 . . 3 (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
216, 12, 16, 20fvmptd 6197 . 2 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
221, 21syl5eq 2656 1 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  {csn 4125  cop 4131   × 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:  elhoma  16505
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