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Theorem elghomlem1 25067
 Description: Lemma for elghom 25069. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
elghomlem1.1
Assertion
Ref Expression
elghomlem1 GrpOpHom
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem elghomlem1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 6716 . . 3
2 rnexg 6716 . . 3
3 elghomlem1.1 . . . 4
43fabexg 6740 . . 3
51, 2, 4syl2an 477 . 2
6 rneq 5228 . . . . . 6
76feq2d 5718 . . . . 5
8 oveq 6290 . . . . . . . . 9
98fveq2d 5870 . . . . . . . 8
109eqeq2d 2481 . . . . . . 7
116, 10raleqbidv 3072 . . . . . 6
126, 11raleqbidv 3072 . . . . 5
137, 12anbi12d 710 . . . 4
1413abbidv 2603 . . 3
15 rneq 5228 . . . . . . 7
16 feq3 5715 . . . . . . 7
1715, 16syl 16 . . . . . 6
18 oveq 6290 . . . . . . . 8
1918eqeq1d 2469 . . . . . . 7
20192ralbidv 2908 . . . . . 6
2117, 20anbi12d 710 . . . . 5
2221abbidv 2603 . . . 4
2322, 3syl6eqr 2526 . . 3
24 df-ghom 25064 . . 3 GrpOpHom
2514, 23, 24ovmpt2g 6421 . 2 GrpOpHom
265, 25mpd3an3 1325 1 GrpOpHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  cab 2452  wral 2814  cvv 3113   crn 5000  wf 5584  cfv 5588  (class class class)co 6284  cgr 24892   GrpOpHom cghom 25063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-ghom 25064 This theorem is referenced by:  elghomlem2  25068
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