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Theorem ghomf 37876
Description: Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
Hypotheses
Ref Expression
ghomf.1 𝑋 = ran 𝐺
ghomf.2 𝑊 = ran 𝐻
Assertion
Ref Expression
ghomf ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋𝑊)

Proof of Theorem ghomf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomf.1 . . . 4 𝑋 = ran 𝐺
2 ghomf.2 . . . 4 𝑊 = ran 𝐻
31, 2elghomOLD 37873 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋𝑊 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
43simprbda 498 . 2 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋𝑊)
543impa 1109 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  ran crn 5689  wf 6558  cfv 6562  (class class class)co 7430  GrpOpcgr 30517   GrpOpHom cghomOLD 37869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-ghomOLD 37870
This theorem is referenced by:  ghomdiv  37878  grpokerinj  37879
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