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Theorem elgiso 25099
 Description: Membership in the set of group isomorphisms from to . (Contributed by Paul Chapman, 25-Feb-2008.)
Assertion
Ref Expression
elgiso GrpOpHom

Proof of Theorem elgiso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6080 . . . . 5 GrpOpHom GrpOpHom
2 rneq 5087 . . . . . 6
3 f1oeq2 5658 . . . . . 6
42, 3syl 16 . . . . 5
51, 4rabeqbidv 2943 . . . 4 GrpOpHom GrpOpHom
6 oveq2 6081 . . . . 5 GrpOpHom GrpOpHom
7 rneq 5087 . . . . . 6
8 f1oeq3 5659 . . . . . 6
97, 8syl 16 . . . . 5
106, 9rabeqbidv 2943 . . . 4 GrpOpHom GrpOpHom
11 df-giso 21940 . . . 4 GrpOpHom
12 ovex 6098 . . . . 5 GrpOpHom
1312rabex 4346 . . . 4 GrpOpHom
145, 10, 11, 13ovmpt2 6201 . . 3 GrpOpHom
1514eleq2d 2502 . 2 GrpOpHom
16 f1oeq1 5657 . . 3
1716elrab 3084 . 2 GrpOpHom GrpOpHom
1815, 17syl6bb 253 1 GrpOpHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  crab 2701   crn 4871  wf1o 5445  (class class class)co 6073  cgr 21766   GrpOpHom cghom 21937   cgiso 21939 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-giso 21940
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