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Theorem grpoinvf 24906
 Description: Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpasscan1.1
grpasscan1.2
Assertion
Ref Expression
grpoinvf

Proof of Theorem grpoinvf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6242 . . . 4 GId
2 eqid 2462 . . . 4 GId GId
31, 2fnmpti 5702 . . 3 GId
4 grpasscan1.1 . . . . 5
5 eqid 2462 . . . . 5 GId GId
6 grpasscan1.2 . . . . 5
74, 5, 6grpoinvfval 24890 . . . 4 GId
87fneq1d 5664 . . 3 GId
93, 8mpbiri 233 . 2
10 fnrnfv 5907 . . . 4
119, 10syl 16 . . 3
124, 6grpoinvcl 24892 . . . . . . 7
134, 6grpo2inv 24905 . . . . . . . 8
1413eqcomd 2470 . . . . . . 7
15 fveq2 5859 . . . . . . . . 9
1615eqeq2d 2476 . . . . . . . 8
1716rspcev 3209 . . . . . . 7
1812, 14, 17syl2anc 661 . . . . . 6
1918ex 434 . . . . 5
20 simpr 461 . . . . . . . 8
214, 6grpoinvcl 24892 . . . . . . . . 9
2221adantr 465 . . . . . . . 8
2320, 22eqeltrd 2550 . . . . . . 7
2423exp31 604 . . . . . 6
2524rexlimdv 2948 . . . . 5
2619, 25impbid 191 . . . 4
2726abbi2dv 2599 . . 3
2811, 27eqtr4d 2506 . 2
29 fveq2 5859 . . . 4
304, 6grpo2inv 24905 . . . . . 6
3130, 13eqeqan12d 2485 . . . . 5
3231anandis 827 . . . 4
3329, 32syl5ib 219 . . 3
3433ralrimivva 2880 . 2
35 dff1o6 6162 . 2
369, 28, 34, 35syl3anbrc 1175 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1374   wcel 1762  cab 2447  wral 2809  wrex 2810   cmpt 4500   crn 4995   wfn 5576  wf1o 5580  cfv 5581  crio 6237  (class class class)co 6277  cgr 24852  GIdcgi 24853  cgn 24854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-grpo 24857  df-gid 24858  df-ginv 24859 This theorem is referenced by:  ginvsn  25015  nvinvfval  25199
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