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Theorem homfval 17637
Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
homffval.f 𝐹 = (Homf𝐶)
homffval.b 𝐵 = (Base‘𝐶)
homffval.h 𝐻 = (Hom ‘𝐶)
homfval.x (𝜑𝑋𝐵)
homfval.y (𝜑𝑌𝐵)
Assertion
Ref Expression
homfval (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))

Proof of Theorem homfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homffval.f . . . 4 𝐹 = (Homf𝐶)
2 homffval.b . . . 4 𝐵 = (Base‘𝐶)
3 homffval.h . . . 4 𝐻 = (Hom ‘𝐶)
41, 2, 3homffval 17635 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
54a1i 11 . 2 (𝜑𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
6 oveq12 7411 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
76adantl 481 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
8 homfval.x . 2 (𝜑𝑋𝐵)
9 homfval.y . 2 (𝜑𝑌𝐵)
10 ovexd 7437 . 2 (𝜑 → (𝑋𝐻𝑌) ∈ V)
115, 7, 8, 9, 10ovmpod 7553 1 (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cfv 6534  (class class class)co 7402  cmpo 7404  Basecbs 17145  Hom chom 17209  Homf chomf 17611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-homf 17615
This theorem is referenced by:  homfeqval  17642  comfffval2  17646  comffval2  17647  comfval2  17648  catsubcat  17790  subcss2  17794  fullsubc  17801  fullresc  17802  funcres2c  17855  hof1  18211  hofcllem  18215  hofcl  18216  yonffthlem  18239  srhmsubc  20568  srhmsubcALTV  47213
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