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Theorem homaval 17266
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 7132 . 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 17264 . . 3 (𝜑𝐻 = (𝑧 ∈ (𝐵 × 𝐵) ↦ ({𝑧} × (𝐽𝑧))))
7 simpr 487 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
87sneqd 4551 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → {𝑧} = {⟨𝑋, 𝑌⟩})
97fveq2d 6646 . . . . 5 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝐽‘⟨𝑋, 𝑌⟩))
10 df-ov 7132 . . . . 5 (𝑋𝐽𝑌) = (𝐽‘⟨𝑋, 𝑌⟩)
119, 10syl6eqr 2873 . . . 4 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → (𝐽𝑧) = (𝑋𝐽𝑌))
128, 11xpeq12d 5558 . . 3 ((𝜑𝑧 = ⟨𝑋, 𝑌⟩) → ({𝑧} × (𝐽𝑧)) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
13 homaval.x . . . 4 (𝜑𝑋𝐵)
14 homaval.y . . . 4 (𝜑𝑌𝐵)
1513, 14opelxpd 5565 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
16 snex 5304 . . . . 5 {⟨𝑋, 𝑌⟩} ∈ V
17 ovex 7162 . . . . 5 (𝑋𝐽𝑌) ∈ V
1816, 17xpex 7450 . . . 4 ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V
1918a1i 11 . . 3 (𝜑 → ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)) ∈ V)
206, 12, 15, 19fvmptd 6747 . 2 (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
211, 20syl5eq 2867 1 (𝜑 → (𝑋𝐻𝑌) = ({⟨𝑋, 𝑌⟩} × (𝑋𝐽𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  Vcvv 3470  {csn 4539  cop 4545   × cxp 5525  cfv 6327  (class class class)co 7129  Basecbs 16458  Hom chom 16551  Catccat 16910  Homachoma 17258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-iun 4893  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7132  df-homa 17261
This theorem is referenced by:  elhoma  17267
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