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Theorem elghomlem1OLD 35316
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 35318. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
elghomlem1OLD.1 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}
Assertion
Ref Expression
elghomlem1OLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐺   𝑓,𝐻,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑓)

Proof of Theorem elghomlem1OLD
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 7599 . . 3 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2 rnexg 7599 . . 3 (𝐻 ∈ GrpOp → ran 𝐻 ∈ V)
3 elghomlem1OLD.1 . . . 4 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}
43fabexg 7625 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
51, 2, 4syl2an 598 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6 rneq 5774 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
76feq2d 6477 . . . . 5 (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran 𝑓:ran 𝐺⟶ran ))
8 oveq 7145 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
98fveq2d 6653 . . . . . . . 8 (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
109eqeq2d 2812 . . . . . . 7 (𝑔 = 𝐺 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
116, 10raleqbidv 3357 . . . . . 6 (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
126, 11raleqbidv 3357 . . . . 5 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
137, 12anbi12d 633 . . . 4 (𝑔 = 𝐺 → ((𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦))) ↔ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))))
1413abbidv 2865 . . 3 (𝑔 = 𝐺 → {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))})
15 rneq 5774 . . . . . . 7 ( = 𝐻 → ran = ran 𝐻)
1615feq3d 6478 . . . . . 6 ( = 𝐻 → (𝑓:ran 𝐺⟶ran 𝑓:ran 𝐺⟶ran 𝐻))
17 oveq 7145 . . . . . . . 8 ( = 𝐻 → ((𝑓𝑥)(𝑓𝑦)) = ((𝑓𝑥)𝐻(𝑓𝑦)))
1817eqeq1d 2803 . . . . . . 7 ( = 𝐻 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19182ralbidv 3167 . . . . . 6 ( = 𝐻 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
2016, 19anbi12d 633 . . . . 5 ( = 𝐻 → ((𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))) ↔ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))))
2120abbidv 2865 . . . 4 ( = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))})
2221, 3eqtr4di 2854 . . 3 ( = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = 𝑆)
23 df-ghomOLD 35315 . . 3 GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
2414, 22, 23ovmpog 7292 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1459 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  Vcvv 3444  ran crn 5524  wf 6324  cfv 6328  (class class class)co 7139  GrpOpcgr 28275   GrpOpHom cghomOLD 35314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-ghomOLD 35315
This theorem is referenced by:  elghomlem2OLD  35317
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