Theorem rnghmf 20361

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 20360	. 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 19127	. 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 2111  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  Basecbs 17115   GrpHom cghm 19119   RngHom crnghm 20347

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-map 8747  df-ghm 19120  df-abl 19690  df-rng 20066  df-rnghm 20349

This theorem is referenced by:  rnghmf1o  20365  rngimcnv  20369  elrngchom  20534  rnghmsscmap2  20539  rnghmsscmap  20540  rnghmsubcsetclem2  20542  rngcsect  20546  rngcinv  20547  funcrngcsetc  20550  funcrngcsetcALT  20551  zrinitorngc  20552  zrtermorngc  20553  elrngchomALTV  48300  rngcinvALTV  48307