Theorem rnghmf 20684

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 20683	. 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 19440	. 2 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐵⟶𝐶)
5	1, 4	syl 18	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421  Basecbs 17393   GrpHom cghm 19433   RngHom crnghm 20670

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-map 8853  df-ghm 19434  df-abl 20003  df-rng 20381  df-rnghm 20672

This theorem is used by:  rnghmf1o  20688  rngimcnv  20692  elrngchom  20882  rnghmsscmap2  20887  rnghmsscmap  20888  rnghmsubcsetclem2  20890  rngcsect  20894  rngcinv  20895  funcrngcsetc  20898  funcrngcsetcALT  20899  zrinitorngc  20900  zrtermorngc  20901  elrngchomALTV  49460  rngcinvALTV  49467