Theorem rnghmf 20419

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 20418	. 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 19186	. 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 1547   ∈ wcel 2119  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Basecbs 17170   GrpHom cghm 19178   RngHom crnghm 20405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-ghm 19179  df-abl 19749  df-rng 20125  df-rnghm 20407

This theorem is referenced by:  rnghmf1o  20423  rngimcnv  20427  elrngchom  20596  rnghmsscmap2  20601  rnghmsscmap  20602  rnghmsubcsetclem2  20604  rngcsect  20608  rngcinv  20609  funcrngcsetc  20612  funcrngcsetcALT  20613  zrinitorngc  20614  zrtermorngc  20615  elrngchomALTV  48760  rngcinvALTV  48767