Theorem rnghmf 20428

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 20427	. 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 19195	. 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 1542   ∈ wcel 2114  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   GrpHom cghm 19187   RngHom crnghm 20414

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ghm 19188  df-abl 19758  df-rng 20134  df-rnghm 20416

This theorem is referenced by:  rnghmf1o  20432  rngimcnv  20436  elrngchom  20601  rnghmsscmap2  20606  rnghmsscmap  20607  rnghmsubcsetclem2  20609  rngcsect  20613  rngcinv  20614  funcrngcsetc  20617  funcrngcsetcALT  20618  zrinitorngc  20619  zrtermorngc  20620  elrngchomALTV  48745  rngcinvALTV  48752