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Theorem rnghmf 20387
Description: A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.)
Hypotheses
Ref Expression
rnghmf.b 𝐵 = (Base‘𝑅)
rnghmf.c 𝐶 = (Base‘𝑆)
Assertion
Ref Expression
rnghmf (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵𝐶)

Proof of Theorem rnghmf
StepHypRef Expression
1 rnghmghm 20386 . 2 (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2 rnghmf.b . . 3 𝐵 = (Base‘𝑅)
3 rnghmf.c . . 3 𝐶 = (Base‘𝑆)
42, 3ghmf 19174 . 2 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵𝐶)
51, 4syl 17 1 (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wf 6544  cfv 6548  (class class class)co 7420  Basecbs 17180   GrpHom cghm 19167   RngHom crnghm 20373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-ghm 19168  df-abl 19738  df-rng 20093  df-rnghm 20375
This theorem is referenced by:  rnghmf1o  20391  rngimcnv  20395  elrngchom  20557  rnghmsscmap2  20562  rnghmsscmap  20563  rnghmsubcsetclem2  20565  rngcsect  20569  rngcinv  20570  funcrngcsetc  20573  funcrngcsetcALT  20574  zrinitorngc  20575  zrtermorngc  20576  elrngchomALTV  47331  rngcinvALTV  47338
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