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Theorem homfval 16962
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 16960 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
54a1i 11 . 2 (𝜑𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
6 oveq12 7158 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
76adantl 485 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
8 homfval.x . 2 (𝜑𝑋𝐵)
9 homfval.y . 2 (𝜑𝑌𝐵)
10 ovexd 7184 . 2 (𝜑 → (𝑋𝐻𝑌) ∈ V)
115, 7, 8, 9, 10ovmpod 7295 1 (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cfv 6343  (class class class)co 7149  cmpo 7151  Basecbs 16483  Hom chom 16576  Homf chomf 16937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-homf 16941
This theorem is referenced by:  homfeqval  16967  comfffval2  16971  comffval2  16972  comfval2  16973  catsubcat  17109  subcss2  17113  fullsubc  17120  fullresc  17121  funcres2c  17171  hof1  17504  hofcllem  17508  hofcl  17509  yonffthlem  17532  srhmsubc  44631  srhmsubcALTV  44649
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