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Theorem elghomlem2OLD 37058
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 37059. (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
elghomlem2OLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐹   𝑓,𝐺,π‘₯,𝑦   𝑓,𝐻,π‘₯,𝑦
Allowed substitution hints:   𝑆(π‘₯,𝑦,𝑓)

Proof of Theorem elghomlem2OLD
StepHypRef Expression
1 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)))}
21elghomlem1OLD 37057 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
32eleq2d 2818 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ 𝐹 ∈ 𝑆))
4 elex 3492 . . . . 5 (𝐹 ∈ 𝑆 β†’ 𝐹 ∈ V)
5 feq1 6699 . . . . . . . 8 (𝑓 = 𝐹 β†’ (𝑓:ran 𝐺⟢ran 𝐻 ↔ 𝐹:ran 𝐺⟢ran 𝐻))
6 fveq1 6891 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
7 fveq1 6891 . . . . . . . . . . 11 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
86, 7oveq12d 7430 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)))
9 fveq1 6891 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯𝐺𝑦)) = (πΉβ€˜(π‘₯𝐺𝑦)))
108, 9eqeq12d 2747 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)) ↔ ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))))
11102ralbidv 3217 . . . . . . . 8 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦)) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))))
125, 11anbi12d 630 . . . . . . 7 (𝑓 = 𝐹 β†’ ((𝑓:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘“β€˜π‘₯)𝐻(π‘“β€˜π‘¦)) = (π‘“β€˜(π‘₯𝐺𝑦))) ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
1312, 1elab2g 3671 . . . . . 6 (𝐹 ∈ V β†’ (𝐹 ∈ 𝑆 ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
1413biimpd 228 . . . . 5 (𝐹 ∈ V β†’ (𝐹 ∈ 𝑆 β†’ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
154, 14mpcom 38 . . . 4 (𝐹 ∈ 𝑆 β†’ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))))
16 rnexg 7898 . . . . . . 7 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
17 fex 7231 . . . . . . . 8 ((𝐹:ran 𝐺⟢ran 𝐻 ∧ ran 𝐺 ∈ V) β†’ 𝐹 ∈ V)
1817expcom 413 . . . . . . 7 (ran 𝐺 ∈ V β†’ (𝐹:ran 𝐺⟢ran 𝐻 β†’ 𝐹 ∈ V))
1916, 18syl 17 . . . . . 6 (𝐺 ∈ GrpOp β†’ (𝐹:ran 𝐺⟢ran 𝐻 β†’ 𝐹 ∈ V))
2019adantrd 491 . . . . 5 (𝐺 ∈ GrpOp β†’ ((𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))) β†’ 𝐹 ∈ V))
2113biimprd 247 . . . . 5 (𝐹 ∈ V β†’ ((𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))) β†’ 𝐹 ∈ 𝑆))
2220, 21syli 39 . . . 4 (𝐺 ∈ GrpOp β†’ ((𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦))) β†’ 𝐹 ∈ 𝑆))
2315, 22impbid2 225 . . 3 (𝐺 ∈ GrpOp β†’ (𝐹 ∈ 𝑆 ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
2423adantr 480 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ 𝑆 ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
253, 24bitrd 278 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟢ran 𝐻 ∧ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  Vcvv 3473  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412  GrpOpcgr 30006   GrpOpHom cghomOLD 37055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ghomOLD 37056
This theorem is referenced by:  elghomOLD  37059
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