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Definition df-ghomOLD 32853
 Description: Obsolete version of df-ghm 17481 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to ℎ. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Assertion
Ref Expression
df-ghomOLD GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
Distinct variable group:   𝑓,𝑔,,𝑥,𝑦

Detailed syntax breakdown of Definition df-ghomOLD
StepHypRef Expression
1 cghomOLD 32852 . 2 class GrpOpHom
2 vg . . 3 setvar 𝑔
3 vh . . 3 setvar
4 cgr 26727 . . 3 class GrpOp
52cv 1474 . . . . . . 7 class 𝑔
65crn 5039 . . . . . 6 class ran 𝑔
73cv 1474 . . . . . . 7 class
87crn 5039 . . . . . 6 class ran
9 vf . . . . . . 7 setvar 𝑓
109cv 1474 . . . . . 6 class 𝑓
116, 8, 10wf 5800 . . . . 5 wff 𝑓:ran 𝑔⟶ran
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1474 . . . . . . . . . 10 class 𝑥
1413, 10cfv 5804 . . . . . . . . 9 class (𝑓𝑥)
15 vy . . . . . . . . . . 11 setvar 𝑦
1615cv 1474 . . . . . . . . . 10 class 𝑦
1716, 10cfv 5804 . . . . . . . . 9 class (𝑓𝑦)
1814, 17, 7co 6549 . . . . . . . 8 class ((𝑓𝑥)(𝑓𝑦))
1913, 16, 5co 6549 . . . . . . . . 9 class (𝑥𝑔𝑦)
2019, 10cfv 5804 . . . . . . . 8 class (𝑓‘(𝑥𝑔𝑦))
2118, 20wceq 1475 . . . . . . 7 wff ((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))
2221, 15, 6wral 2896 . . . . . 6 wff 𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))
2322, 12, 6wral 2896 . . . . 5 wff 𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))
2411, 23wa 383 . . . 4 wff (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))
2524, 9cab 2596 . . 3 class {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))}
262, 3, 4, 4, 25cmpt2 6551 . 2 class (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
271, 26wceq 1475 1 wff GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
 Colors of variables: wff setvar class This definition is referenced by:  elghomlem1OLD  32854
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