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Theorem rngohomf 36428
Description: A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
rnghomf.1 𝐺 = (1st𝑅)
rnghomf.2 𝑋 = ran 𝐺
rnghomf.3 𝐽 = (1st𝑆)
rnghomf.4 𝑌 = ran 𝐽
Assertion
Ref Expression
rngohomf ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋𝑌)

Proof of Theorem rngohomf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghomf.1 . . . . 5 𝐺 = (1st𝑅)
2 eqid 2737 . . . . 5 (2nd𝑅) = (2nd𝑅)
3 rnghomf.2 . . . . 5 𝑋 = ran 𝐺
4 eqid 2737 . . . . 5 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
5 rnghomf.3 . . . . 5 𝐽 = (1st𝑆)
6 eqid 2737 . . . . 5 (2nd𝑆) = (2nd𝑆)
7 rnghomf.4 . . . . 5 𝑌 = ran 𝐽
8 eqid 2737 . . . . 5 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
91, 2, 3, 4, 5, 6, 7, 8isrngohom 36427 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋𝑌 ∧ (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))))
109biimpa 478 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋𝑌 ∧ (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))))
1110simp1d 1143 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋𝑌)
12113impa 1111 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  ran crn 5635  wf 6493  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356   RngHom crnghom 36422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-rngohom 36425
This theorem is referenced by:  rngohomcl  36429  rngogrphom  36433  rngohomco  36436  keridl  36494
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