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Theorem homfval 16292
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 16290 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
54a1i 11 . 2 (𝜑𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
6 oveq12 6624 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
76adantl 482 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
8 homfval.x . 2 (𝜑𝑋𝐵)
9 homfval.y . 2 (𝜑𝑌𝐵)
10 ovexd 6645 . 2 (𝜑 → (𝑋𝐻𝑌) ∈ V)
115, 7, 8, 9, 10ovmpt2d 6753 1 (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cfv 5857  (class class class)co 6615  cmpt2 6617  Basecbs 15800  Hom chom 15892  Homf chomf 16267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-homf 16271
This theorem is referenced by:  homfeqval  16297  comfffval2  16301  comffval2  16302  comfval2  16303  catsubcat  16439  subcss2  16443  fullsubc  16450  fullresc  16451  funcres2c  16501  hof1  16834  hofcllem  16838  hofcl  16839  yonffthlem  16862  srhmsubc  41394  srhmsubcALTV  41412
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