Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-ghomOLD Structured version   Visualization version   GIF version

Definition df-ghomOLD 32636
Description: Obsolete version of df-ghm 17429 as of 15-Mar-2020. Define the set of group homomorphisms from 𝑔 to . (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 32635 . 2 class GrpOpHom
2 vg . . 3 setvar 𝑔
3 vh . . 3 setvar
4 cgr 26520 . . 3 class GrpOp
52cv 1473 . . . . . . 7 class 𝑔
65crn 5028 . . . . . 6 class ran 𝑔
73cv 1473 . . . . . . 7 class
87crn 5028 . . . . . 6 class ran
9 vf . . . . . . 7 setvar 𝑓
109cv 1473 . . . . . 6 class 𝑓
116, 8, 10wf 5785 . . . . 5 wff 𝑓:ran 𝑔⟶ran
12 vx . . . . . . . . . . 11 setvar 𝑥
1312cv 1473 . . . . . . . . . 10 class 𝑥
1413, 10cfv 5789 . . . . . . . . 9 class (𝑓𝑥)
15 vy . . . . . . . . . . 11 setvar 𝑦
1615cv 1473 . . . . . . . . . 10 class 𝑦
1716, 10cfv 5789 . . . . . . . . 9 class (𝑓𝑦)
1814, 17, 7co 6526 . . . . . . . 8 class ((𝑓𝑥)(𝑓𝑦))
1913, 16, 5co 6526 . . . . . . . . 9 class (𝑥𝑔𝑦)
2019, 10cfv 5789 . . . . . . . 8 class (𝑓‘(𝑥𝑔𝑦))
2118, 20wceq 1474 . . . . . . 7 wff ((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))
2221, 15, 6wral 2895 . . . . . 6 wff 𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))
2322, 12, 6wral 2895 . . . . 5 wff 𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))
2411, 23wa 382 . . . 4 wff (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))
2524, 9cab 2595 . . 3 class {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))}
262, 3, 4, 4, 25cmpt2 6528 . 2 class (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
271, 26wceq 1474 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  32637
  Copyright terms: Public domain W3C validator