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Theorem rnghmf 46283
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 RngHomo 𝑆) → 𝐹:𝐵𝐶)

Proof of Theorem rnghmf
StepHypRef Expression
1 rnghmghm 46282 . 2 (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2 rnghmf.b . . 3 𝐵 = (Base‘𝑅)
3 rnghmf.c . . 3 𝐶 = (Base‘𝑆)
42, 3ghmf 19017 . 2 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵𝐶)
51, 4syl 17 1 (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹:𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17088   GrpHom cghm 19010   RngHomo crngh 46269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-ghm 19011  df-abl 19570  df-rng 46259  df-rnghomo 46271
This theorem is referenced by:  rnghmf1o  46287  elrngchom  46352  rnghmsscmap2  46357  rnghmsscmap  46358  rnghmsubcsetclem2  46360  rngcsect  46364  rngcinv  46365  elrngchomALTV  46370  rngcinvALTV  46377  funcrngcsetc  46382  funcrngcsetcALT  46383  zrinitorngc  46384  zrtermorngc  46385
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