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Theorem elghomlem1OLD 33997
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 33999. (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 7263 . . 3 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2 rnexg 7263 . . 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 7287 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
51, 2, 4syl2an 495 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6 rneq 5506 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
76feq2d 6192 . . . . 5 (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran 𝑓:ran 𝐺⟶ran ))
8 oveq 6819 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
98fveq2d 6356 . . . . . . . 8 (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
109eqeq2d 2770 . . . . . . 7 (𝑔 = 𝐺 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
116, 10raleqbidv 3291 . . . . . 6 (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
126, 11raleqbidv 3291 . . . . 5 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
137, 12anbi12d 749 . . . 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 2879 . . 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 5506 . . . . . . 7 ( = 𝐻 → ran = ran 𝐻)
1615feq3d 6193 . . . . . 6 ( = 𝐻 → (𝑓:ran 𝐺⟶ran 𝑓:ran 𝐺⟶ran 𝐻))
17 oveq 6819 . . . . . . . 8 ( = 𝐻 → ((𝑓𝑥)(𝑓𝑦)) = ((𝑓𝑥)𝐻(𝑓𝑦)))
1817eqeq1d 2762 . . . . . . 7 ( = 𝐻 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19182ralbidv 3127 . . . . . 6 ( = 𝐻 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
2016, 19anbi12d 749 . . . . 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 2879 . . . 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 2812 . . 3 ( = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = 𝑆)
23 df-ghomOLD 33996 . . 3 GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
2414, 22, 23ovmpt2g 6960 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1574 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  Vcvv 3340  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6813  GrpOpcgr 27652   GrpOpHom cghomOLD 33995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-ghomOLD 33996
This theorem is referenced by:  elghomlem2OLD  33998
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