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Theorem ghmf 19238
Description: A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmf.x 𝑋 = (Base‘𝑆)
ghmf.y 𝑌 = (Base‘𝑇)
Assertion
Ref Expression
ghmf (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)

Proof of Theorem ghmf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmf.x . . . 4 𝑋 = (Base‘𝑆)
2 ghmf.y . . . 4 𝑌 = (Base‘𝑇)
3 eqid 2737 . . . 4 (+g𝑆) = (+g𝑆)
4 eqid 2737 . . . 4 (+g𝑇) = (+g𝑇)
51, 2, 3, 4isghm 19233 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦𝑋𝑥𝑋 (𝐹‘(𝑦(+g𝑆)𝑥)) = ((𝐹𝑦)(+g𝑇)(𝐹𝑥)))))
65simprbi 496 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋𝑌 ∧ ∀𝑦𝑋𝑥𝑋 (𝐹‘(𝑦(+g𝑆)𝑥)) = ((𝐹𝑦)(+g𝑇)(𝐹𝑥))))
76simpld 494 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951   GrpHom cghm 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ghm 19231
This theorem is referenced by:  ghmid  19240  ghminv  19241  ghmsub  19242  ghmmhm  19244  ghmmulg  19246  ghmrn  19247  resghm  19250  ghmpreima  19256  ghmeql  19257  ghmnsgima  19258  ghmnsgpreima  19259  ghmeqker  19261  ghmf1  19264  kerf1ghm  19265  ghmf1o  19266  gimcnv  19285  ghmqusnsglem1  19298  ghmqusnsg  19300  ghmquskerlem1  19301  ghmquskerco  19302  ghmquskerlem3  19304  ghmqusker  19305  lactghmga  19423  frgpup3lem  19795  frgpup3  19796  ghmplusg  19864  isrnghmmul  20442  rnghmf  20448  rhmf  20485  isrhm2d  20487  lmhmf  21033  lmhmpropd  21072  frgpcyg  21592  psgninv  21600  zrhpsgninv  21603  evpmss  21604  psgnevpmb  21605  psgnodpm  21606  zrhpsgnevpm  21609  zrhpsgnodpm  21610  evlslem2  22103  nmoi  24749  nmoix  24750  nmoi2  24751  nmoleub  24752  nmoeq0  24757  nmoco  24758  nmotri  24760  nmods  24765  nghmcn  24766  aks6d1c1p2  42110  aks6d1c1p3  42111  aks6d1c5lem1  42137
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