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Theorem homfval 16705
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 16703 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
54a1i 11 . 2 (𝜑𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
6 oveq12 6915 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
76adantl 475 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
8 homfval.x . 2 (𝜑𝑋𝐵)
9 homfval.y . 2 (𝜑𝑌𝐵)
10 ovexd 6940 . 2 (𝜑 → (𝑋𝐻𝑌) ∈ V)
115, 7, 8, 9, 10ovmpt2d 7049 1 (𝜑 → (𝑋𝐹𝑌) = (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  Vcvv 3415  cfv 6124  (class class class)co 6906  cmpt2 6908  Basecbs 16223  Hom chom 16317  Homf chomf 16680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-homf 16684
This theorem is referenced by:  homfeqval  16710  comfffval2  16714  comffval2  16715  comfval2  16716  catsubcat  16852  subcss2  16856  fullsubc  16863  fullresc  16864  funcres2c  16914  hof1  17248  hofcllem  17252  hofcl  17253  yonffthlem  17276  srhmsubc  42924  srhmsubcALTV  42942
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