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Theorem rngohomcl 37481
Description: Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
rnghomf.1 𝐺 = (1st𝑅)
rnghomf.2 𝑋 = ran 𝐺
rnghomf.3 𝐽 = (1st𝑆)
rnghomf.4 𝑌 = ran 𝐽
Assertion
Ref Expression
rngohomcl (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)

Proof of Theorem rngohomcl
StepHypRef Expression
1 rnghomf.1 . . 3 𝐺 = (1st𝑅)
2 rnghomf.2 . . 3 𝑋 = ran 𝐺
3 rnghomf.3 . . 3 𝐽 = (1st𝑆)
4 rnghomf.4 . . 3 𝑌 = ran 𝐽
51, 2, 3, 4rngohomf 37480 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋𝑌)
65ffvelcdmda 7099 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  ran crn 5683  cfv 6553  (class class class)co 7426  1st c1st 7999  RingOpscrngo 37408   RingOpsHom crngohom 37474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8855  df-rngohom 37477
This theorem is referenced by:  rngohomco  37488  keridl  37546
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