Theorem rngohomcl 36119

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 RngHom 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 36118	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋⟶𝑌)
6	5	ffvelrnda 6956	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ran crn 5590  ‘cfv 6431  (class class class)co 7269  1^st c1st 7820  RingOpscrngo 36046   RngHom crnghom 36112

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-map 8598  df-rngohom 36115

This theorem is referenced by:  rngohomco  36126  keridl  36184