Theorem rngohomcl 38506

Description: Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)

Hypotheses

Ref	Expression
rnghomf.1	⊢ 𝐺 = (1st ‘𝑅)
rnghomf.2	⊢ 𝑋 = ran 𝐺
rnghomf.3	⊢ 𝐽 = (1st ‘𝑆)
rnghomf.4	⊢ 𝑌 = ran 𝐽

Assertion

Ref	Expression
rngohomcl	⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)

Proof of Theorem rngohomcl

Step	Hyp	Ref	Expression
1		rnghomf.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		rnghomf.2	. . 3 ⊢ 𝑋 = ran 𝐺
3		rnghomf.3	. . 3 ⊢ 𝐽 = (1st ‘𝑆)
4		rnghomf.4	. . 3 ⊢ 𝑌 = ran 𝐽
5	1, 2, 3, 4	rngohomf 38505	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)
6	5	ffvelcdmda 7080	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ran crn 5663  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  RingOpscrngo 38433   RingOpsHom crngohom 38499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-rngohom 38502

This theorem is referenced by:  rngohomco  38513  keridl  38571