Theorem ghomf 38961

Description: Obsolete theorem, use ghmf 19443 instead. Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

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:𝑋⟶𝑊)

Proof of Theorem ghomf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghomf.1	. . . 4 ⊢ 𝑋 = ran 𝐺
2		ghomf.2	. . . 4 ⊢ 𝑊 = ran 𝐻
3	1, 2	elghomOLD 38958	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:𝑋⟶𝑊 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐻(𝐹‘𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
4	3	simprbda 504	. 2 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑊)
5	4	3impa 1127	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋⟶𝑊)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ran crn 5652  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  GrpOpcgr 31100   GrpOpHom cghomOLD 38954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-ghomOLD 38955

This theorem is used by:  ghomdiv  38963  grpokerinj  38964