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Theorem elghomlem1OLD 37056
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37058. (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 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))}
Assertion
Ref Expression
elghomlem1OLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐺   𝑓,𝐻,π‘₯,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦,𝑓)

Proof of Theorem elghomlem1OLD
Dummy variables 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 7897 . . 3 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
2 rnexg 7897 . . 3 (𝐻 ∈ GrpOp β†’ ran 𝐻 ∈ V)
3 elghomlem1OLD.1 . . . 4 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))}
43fabexg 7927 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) β†’ 𝑆 ∈ V)
51, 2, 4syl2an 594 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ 𝑆 ∈ V)
6 rneq 5934 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
76feq2d 6702 . . . . 5 (𝑔 = 𝐺 β†’ (𝑓:ran π‘”βŸΆran β„Ž ↔ 𝑓:ran 𝐺⟢ran β„Ž))
8 oveq 7417 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑦) = (π‘₯𝐺𝑦))
98fveq2d 6894 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘“β€˜(π‘₯𝑔𝑦)) = (π‘“β€˜(π‘₯𝐺𝑦)))
109eqeq2d 2741 . . . . . . 7 (𝑔 = 𝐺 β†’ (((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)) ↔ ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
116, 10raleqbidv 3340 . . . . . 6 (𝑔 = 𝐺 β†’ (βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)) ↔ βˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
126, 11raleqbidv 3340 . . . . 5 (𝑔 = 𝐺 β†’ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
137, 12anbi12d 629 . . . 4 (𝑔 = 𝐺 β†’ ((𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦))) ↔ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))))
1413abbidv 2799 . . 3 (𝑔 = 𝐺 β†’ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))})
15 rneq 5934 . . . . . . 7 (β„Ž = 𝐻 β†’ ran β„Ž = ran 𝐻)
1615feq3d 6703 . . . . . 6 (β„Ž = 𝐻 β†’ (𝑓:ran 𝐺⟢ran β„Ž ↔ 𝑓:ran 𝐺⟢ran 𝐻))
17 oveq 7417 . . . . . . . 8 (β„Ž = 𝐻 β†’ ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = ((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)))
1817eqeq1d 2732 . . . . . . 7 (β„Ž = 𝐻 β†’ (((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)) ↔ ((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
19182ralbidv 3216 . . . . . 6 (β„Ž = 𝐻 β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))))
2016, 19anbi12d 629 . . . . 5 (β„Ž = 𝐻 β†’ ((𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))) ↔ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))))
2120abbidv 2799 . . . 4 (β„Ž = 𝐻 β†’ {𝑓 ∣ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))} = {𝑓 ∣ (𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))})
2221, 3eqtr4di 2788 . . 3 (β„Ž = 𝐻 β†’ {𝑓 ∣ (𝑓:ran 𝐺⟢ran β„Ž ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)))} = 𝑆)
23 df-ghomOLD 37055 . . 3 GrpOpHom = (𝑔 ∈ GrpOp, β„Ž ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran π‘”βŸΆran β„Ž ∧ βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝑔𝑦)))})
2414, 22, 23ovmpog 7569 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1460 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  Vcvv 3472  ran crn 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  GrpOpcgr 30009   GrpOpHom cghomOLD 37054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-ghomOLD 37055
This theorem is referenced by:  elghomlem2OLD  37057
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