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Theorem elghomlem2OLD 35781
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 35782. (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
elghomlem2OLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑓,𝐺,𝑥,𝑦   𝑓,𝐻,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑓)

Proof of Theorem elghomlem2OLD
StepHypRef Expression
1 elghomlem1OLD.1 . . . 4 𝑆 = {𝑓 ∣ (𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)))}
21elghomlem1OLD 35780 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐺 GrpOpHom 𝐻) = 𝑆)
32eleq2d 2823 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ 𝐹𝑆))
4 elex 3426 . . . . 5 (𝐹𝑆𝐹 ∈ V)
5 feq1 6526 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓:ran 𝐺⟶ran 𝐻𝐹:ran 𝐺⟶ran 𝐻))
6 fveq1 6716 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
7 fveq1 6716 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
86, 7oveq12d 7231 . . . . . . . . . 10 (𝑓 = 𝐹 → ((𝑓𝑥)𝐻(𝑓𝑦)) = ((𝐹𝑥)𝐻(𝐹𝑦)))
9 fveq1 6716 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓‘(𝑥𝐺𝑦)) = (𝐹‘(𝑥𝐺𝑦)))
108, 9eqeq12d 2753 . . . . . . . . 9 (𝑓 = 𝐹 → (((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
11102ralbidv 3120 . . . . . . . 8 (𝑓 = 𝐹 → (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
125, 11anbi12d 634 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑓𝑥)𝐻(𝑓𝑦)) = (𝑓‘(𝑥𝐺𝑦))) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
1312, 1elab2g 3589 . . . . . 6 (𝐹 ∈ V → (𝐹𝑆 ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
1413biimpd 232 . . . . 5 (𝐹 ∈ V → (𝐹𝑆 → (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
154, 14mpcom 38 . . . 4 (𝐹𝑆 → (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))))
16 rnexg 7682 . . . . . . 7 (𝐺 ∈ GrpOp → ran 𝐺 ∈ V)
17 fex 7042 . . . . . . . 8 ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ran 𝐺 ∈ V) → 𝐹 ∈ V)
1817expcom 417 . . . . . . 7 (ran 𝐺 ∈ V → (𝐹:ran 𝐺⟶ran 𝐻𝐹 ∈ V))
1916, 18syl 17 . . . . . 6 (𝐺 ∈ GrpOp → (𝐹:ran 𝐺⟶ran 𝐻𝐹 ∈ V))
2019adantrd 495 . . . . 5 (𝐺 ∈ GrpOp → ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))) → 𝐹 ∈ V))
2113biimprd 251 . . . . 5 (𝐹 ∈ V → ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))) → 𝐹𝑆))
2220, 21syli 39 . . . 4 (𝐺 ∈ GrpOp → ((𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦))) → 𝐹𝑆))
2315, 22impbid2 229 . . 3 (𝐺 ∈ GrpOp → (𝐹𝑆 ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
2423adantr 484 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹𝑆 ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
253, 24bitrd 282 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) → (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:ran 𝐺⟶ran 𝐻 ∧ ∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝐹𝑥)𝐻(𝐹𝑦)) = (𝐹‘(𝑥𝐺𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  {cab 2714  wral 3061  Vcvv 3408  ran crn 5552  wf 6376  cfv 6380  (class class class)co 7213  GrpOpcgr 28570   GrpOpHom cghomOLD 35778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-ghomOLD 35779
This theorem is referenced by:  elghomOLD  35782
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