Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnghmf Structured version   Visualization version   GIF version

Theorem rnghmf 44453
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 44452 . 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 18362 . 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 1538  wcel 2115  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483   GrpHom cghm 18355   RngHomo crngh 44439
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-ghm 18356  df-abl 18909  df-rng0 44429  df-rnghomo 44441
This theorem is referenced by:  rnghmf1o  44457  elrngchom  44522  rnghmsscmap2  44527  rnghmsscmap  44528  rnghmsubcsetclem2  44530  rngcsect  44534  rngcinv  44535  elrngchomALTV  44540  rngcinvALTV  44547  funcrngcsetc  44552  funcrngcsetcALT  44553  zrinitorngc  44554  zrtermorngc  44555
  Copyright terms: Public domain W3C validator