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Theorem elghomOLD 36396
Description: Obsolete version of isghm 19016 as of 15-Mar-2020. Membership in the set of group homomorphisms from 𝐺 to 𝐻. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
elghomOLD.1 𝑋 = ran 𝐺
elghomOLD.2 π‘Š = ran 𝐻
Assertion
Ref Expression
elghomOLD ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:π‘‹βŸΆπ‘Š ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))))
Distinct variable groups:   π‘₯,𝐹,𝑦   π‘₯,𝐺,𝑦   π‘₯,𝐻,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘Š(π‘₯,𝑦)

Proof of Theorem elghomOLD
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 {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)))} = {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)))}
21elghomlem2OLD 36395 . 2 ((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)))))
3 elghomOLD.1 . . . 4 𝑋 = ran 𝐺
4 elghomOLD.2 . . . 4 π‘Š = ran 𝐻
53, 4feq23i 6666 . . 3 (𝐹:π‘‹βŸΆπ‘Š ↔ 𝐹:ran 𝐺⟢ran 𝐻)
63raleqi 3310 . . . 4 (βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)) ↔ βˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))
73, 6raleqbii 3314 . . 3 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)) ↔ βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(π‘₯𝐺𝑦)))
85, 7anbi12i 628 . 2 ((𝐹:π‘‹βŸΆπ‘Š ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((πΉβ€˜π‘₯)𝐻(πΉβ€˜π‘¦)) = (πΉβ€˜(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))))
92, 8bitr4di 289 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp) β†’ (𝐹 ∈ (𝐺 GrpOpHom 𝐻) ↔ (𝐹:π‘‹βŸΆπ‘Š ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((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 3061  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-rep 5246  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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  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-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-ghomOLD 36393
This theorem is referenced by:  ghomlinOLD  36397  ghomidOLD  36398  ghomf  36399  ghomco  36400  rngogrphom  36480
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