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Theorem elghomlem1OLD 36394
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 36396. (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 7845 . . 3 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
2 rnexg 7845 . . 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 7875 . . 3 ((ran 𝐺 ∈ V ∧ ran 𝐻 ∈ V) β†’ 𝑆 ∈ V)
51, 2, 4syl2an 597 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ 𝑆 ∈ V)
6 rneq 5895 . . . . . 6 (𝑔 = 𝐺 β†’ ran 𝑔 = ran 𝐺)
76feq2d 6658 . . . . 5 (𝑔 = 𝐺 β†’ (𝑓:ran π‘”βŸΆran β„Ž ↔ 𝑓:ran 𝐺⟢ran β„Ž))
8 oveq 7367 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘₯𝑔𝑦) = (π‘₯𝐺𝑦))
98fveq2d 6850 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘“β€˜(π‘₯𝑔𝑦)) = (π‘“β€˜(π‘₯𝐺𝑦)))
109eqeq2d 2744 . . . . . . 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 3318 . . . . . 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 3318 . . . . 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 632 . . . 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 2802 . . 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 5895 . . . . . . 7 (β„Ž = 𝐻 β†’ ran β„Ž = ran 𝐻)
1615feq3d 6659 . . . . . 6 (β„Ž = 𝐻 β†’ (𝑓:ran 𝐺⟢ran β„Ž ↔ 𝑓:ran 𝐺⟢ran 𝐻))
17 oveq 7367 . . . . . . . 8 (β„Ž = 𝐻 β†’ ((π‘“β€˜π‘₯)β„Ž(π‘“β€˜π‘¦)) = ((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)))
1817eqeq1d 2735 . . . . . . 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 3209 . . . . . 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 632 . . . . 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 2802 . . . 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 2791 . . 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 36393 . . 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 7518 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝑆 ∈ V) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
255, 24mpd3an3 1463 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3447  ran crn 5638  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  GrpOpcgr 29480   GrpOpHom cghomOLD 36392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-ghomOLD 36393
This theorem is referenced by:  elghomlem2OLD  36395
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