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Theorem homfval 16175
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 16173 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
54a1i 11 . 2 (𝜑𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
6 oveq12 6558 . . 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 ovex 6577 . . 3 (𝑋𝐻𝑌) ∈ V
1110a1i 11 . 2 (𝜑 → (𝑋𝐻𝑌) ∈ V)
125, 7, 8, 9, 11ovmpt2d 6686 1 (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cfv 5804  (class class class)co 6549  cmpt2 6551  Basecbs 15695  Hom chom 15779  Homf chomf 16150
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-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-homf 16154
This theorem is referenced by:  homfeqval  16180  comfffval2  16184  comffval2  16185  comfval2  16186  catsubcat  16322  subcss2  16326  fullsubc  16333  fullresc  16334  funcres2c  16384  hof1  16717  hofcllem  16721  hofcl  16722  yonffthlem  16745  srhmsubc  41868  srhmsubcALTV  41887
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