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Theorem elghomlem1OLD 32854
 Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 32856. (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 6990 . . 3 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2 rnexg 6990 . . 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 7015 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
51, 2, 4syl2an 493 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6 rneq 5272 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
76feq2d 5944 . . . . 5 (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran 𝑓:ran 𝐺⟶ran ))
8 oveq 6555 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
98fveq2d 6107 . . . . . . . 8 (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
109eqeq2d 2620 . . . . . . 7 (𝑔 = 𝐺 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
116, 10raleqbidv 3129 . . . . . 6 (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
126, 11raleqbidv 3129 . . . . 5 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
137, 12anbi12d 743 . . . 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 2728 . . 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 5272 . . . . . . 7 ( = 𝐻 → ran = ran 𝐻)
1615feq3d 5945 . . . . . 6 ( = 𝐻 → (𝑓:ran 𝐺⟶ran 𝑓:ran 𝐺⟶ran 𝐻))
17 oveq 6555 . . . . . . . 8 ( = 𝐻 → ((𝑓𝑥)(𝑓𝑦)) = ((𝑓𝑥)𝐻(𝑓𝑦)))
1817eqeq1d 2612 . . . . . . 7 ( = 𝐻 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19182ralbidv 2972 . . . . . 6 ( = 𝐻 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
2016, 19anbi12d 743 . . . . 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 2728 . . . 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, 3syl6eqr 2662 . . 3 ( = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = 𝑆)
23 df-ghomOLD 32853 . . 3 GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
2414, 22, 23ovmpt2g 6693 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1417 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  GrpOpcgr 26727   GrpOpHom cghomOLD 32852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ghomOLD 32853 This theorem is referenced by:  elghomlem2OLD  32855
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