Theorem rngohomcl 37954

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 37953	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:𝑋⟶𝑌)
6	5	ffvelcdmda 7038	1 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ran crn 5632  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  RingOpscrngo 37881   RingOpsHom crngohom 37947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-rngohom 37950

This theorem is referenced by:  rngohomco  37961  keridl  38019