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Theorem elghomlem2OLD 36754
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 36755. (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 36753 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐺 GrpOpHom 𝐻) = 𝑆)
32eleq2d 2820 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ 𝐹 ∈ 𝑆))
4 elex 3493 . . . . 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 7427 . . . . . . . . . 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 2749 . . . . . . . . 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 3219 . . . . . . . 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 632 . . . . . . 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 7895 . . . . . . 7 (𝐺 ∈ GrpOp β†’ ran 𝐺 ∈ V)
17 fex 7228 . . . . . . . 8 ((𝐹:ran 𝐺⟢ran 𝐻 ∧ ran 𝐺 ∈ V) β†’ 𝐹 ∈ V)
1817expcom 415 . . . . . . 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 493 . . . . 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 482 . 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 279 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  Vcvv 3475  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  GrpOpcgr 29742   GrpOpHom cghomOLD 36751
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 7412  df-oprab 7413  df-mpo 7414  df-ghomOLD 36752
This theorem is referenced by:  elghomOLD  36755
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