Theorem rnghmf 20375

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:𝐵⟶𝐶)

Proof of Theorem rnghmf

Step	Hyp	Ref	Expression
1		rnghmghm 20374	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2		rnghmf.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rnghmf.c	. . 3 ⊢ 𝐶 = (Base‘𝑆)
4	2, 3	ghmf 19140	. 2 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵⟶𝐶)
5	1, 4	syl 17	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  Basecbs 17127   GrpHom cghm 19132   RngHom crnghm 20361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-map 8761  df-ghm 19133  df-abl 19703  df-rng 20079  df-rnghm 20363

This theorem is referenced by:  rnghmf1o  20379  rngimcnv  20383  elrngchom  20548  rnghmsscmap2  20553  rnghmsscmap  20554  rnghmsubcsetclem2  20556  rngcsect  20560  rngcinv  20561  funcrngcsetc  20564  funcrngcsetcALT  20565  zrinitorngc  20566  zrtermorngc  20567  elrngchomALTV  48431  rngcinvALTV  48438