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Theorem fnhomeqhomf 17644
Description: If the Hom-set operation is a function it is equal to the corresponding functionalized Hom-set operation. (Contributed by AV, 1-Mar-2020.)
Hypotheses
Ref Expression
homffval.f 𝐹 = (Homf𝐶)
homffval.b 𝐵 = (Base‘𝐶)
homffval.h 𝐻 = (Hom ‘𝐶)
Assertion
Ref Expression
fnhomeqhomf (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)

Proof of Theorem fnhomeqhomf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fnov 7536 . 2 (𝐻 Fn (𝐵 × 𝐵) ↔ 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)))
2 homffval.f . . . 4 𝐹 = (Homf𝐶)
3 homffval.b . . . 4 𝐵 = (Base‘𝐶)
4 homffval.h . . . 4 𝐻 = (Hom ‘𝐶)
52, 3, 4homffval 17643 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
6 eqeq2 2738 . . 3 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))))
75, 6mpbiri 258 . 2 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → 𝐹 = 𝐻)
81, 7sylbi 216 1 (𝐻 Fn (𝐵 × 𝐵) → 𝐹 = 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   × cxp 5667   Fn wfn 6532  cfv 6537  (class class class)co 7405  cmpo 7407  Basecbs 17153  Hom chom 17217  Homf chomf 17619
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-homf 17623
This theorem is referenced by:  estrchomfeqhom  18099  rngchomfeqhom  20521  ringchomfeqhom  20550  rngchomffvalALTV  47233
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