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Theorem elghomlem1OLD 33990
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 33992. (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 7324 . . 3 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
2 rnexg 7324 . . 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 7348 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) → 𝑆 ∈ V)
51, 2, 4syl2an 585 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → 𝑆 ∈ V)
6 rneq 5552 . . . . . 6 (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺)
76feq2d 6238 . . . . 5 (𝑔 = 𝐺 → (𝑓:ran 𝑔⟶ran 𝑓:ran 𝐺⟶ran ))
8 oveq 6876 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
98fveq2d 6408 . . . . . . . 8 (𝑔 = 𝐺 → (𝑓‘(𝑥𝑔𝑦)) = (𝑓‘(𝑥𝐺𝑦)))
109eqeq2d 2816 . . . . . . 7 (𝑔 = 𝐺 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
116, 10raleqbidv 3341 . . . . . 6 (𝑔 = 𝐺 → (∀𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
126, 11raleqbidv 3341 . . . . 5 (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
137, 12anbi12d 618 . . . 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 2925 . . 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 5552 . . . . . . 7 ( = 𝐻 → ran = ran 𝐻)
1615feq3d 6239 . . . . . 6 ( = 𝐻 → (𝑓:ran 𝐺⟶ran 𝑓:ran 𝐺⟶ran 𝐻))
17 oveq 6876 . . . . . . . 8 ( = 𝐻 → ((𝑓𝑥)(𝑓𝑦)) = ((𝑓𝑥)𝐻(𝑓𝑦)))
1817eqeq1d 2808 . . . . . . 7 ( = 𝐻 → (((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
19182ralbidv 3177 . . . . . 6 ( = 𝐻 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))))
2016, 19anbi12d 618 . . . . 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 2925 . . . 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 2858 . . 3 ( = 𝐻 → {𝑓 ∣ (𝑓:ran 𝐺⟶ran ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))} = 𝑆)
23 df-ghomOLD 33989 . . 3 GrpOpHom = (𝑔 ∈ GrpOp, ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ∧ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑓𝑥)(𝑓𝑦)) = (𝑓‘(𝑥𝑔𝑦)))})
2414, 22, 23ovmpt2g 7021 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) → (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1579 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  {cab 2792  wral 3096  Vcvv 3391  ran crn 5312  wf 6093  cfv 6097  (class class class)co 6870  GrpOpcgr 27668   GrpOpHom cghomOLD 33988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-ghomOLD 33989
This theorem is referenced by:  elghomlem2OLD  33991
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