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Theorem fnhomeqhomf 17571
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 7487 . 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 17570 . . 3 𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
6 eqeq2 2748 . . 3 (𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦)) → (𝐹 = 𝐻𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))))
75, 6mpbiri 257 . 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 1541   × cxp 5631   Fn wfn 6491  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  Hom chom 17144  Homf chomf 17546
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-homf 17550
This theorem is referenced by:  estrchomfeqhom  18023  rngchomfeqhom  46257  rngchomffvalALTV  46283  ringchomfeqhom  46303
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