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Theorem grpoinvf 21828
 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 6553 . . . 4 GId
2 eqid 2436 . . . 4 GId GId
31, 2fnmpti 5573 . . 3 GId
4 grpasscan1.1 . . . . 5
5 eqid 2436 . . . . 5 GId GId
6 grpasscan1.2 . . . . 5
74, 5, 6grpoinvfval 21812 . . . 4 GId
87fneq1d 5536 . . 3 GId
93, 8mpbiri 225 . 2
10 fnrnfv 5773 . . . 4
119, 10syl 16 . . 3
124, 6grpoinvcl 21814 . . . . . . 7
134, 6grpo2inv 21827 . . . . . . . 8
1413eqcomd 2441 . . . . . . 7
15 fveq2 5728 . . . . . . . . 9
1615eqeq2d 2447 . . . . . . . 8
1716rspcev 3052 . . . . . . 7
1812, 14, 17syl2anc 643 . . . . . 6
1918ex 424 . . . . 5
20 simpr 448 . . . . . . . 8
214, 6grpoinvcl 21814 . . . . . . . . 9
2221adantr 452 . . . . . . . 8
2320, 22eqeltrd 2510 . . . . . . 7
2423exp31 588 . . . . . 6
2524rexlimdv 2829 . . . . 5
2619, 25impbid 184 . . . 4
2726abbi2dv 2551 . . 3
2811, 27eqtr4d 2471 . 2
29 fveq2 5728 . . . 4
304, 6grpo2inv 21827 . . . . . 6
3130, 13eqeqan12d 2451 . . . . 5
3231anandis 804 . . . 4
3329, 32syl5ib 211 . . 3
3433ralrimivva 2798 . 2
35 dff1o6 6013 . 2
369, 28, 34, 35syl3anbrc 1138 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2422  wral 2705  wrex 2706   cmpt 4266   crn 4879   wfn 5449  wf1o 5453  cfv 5454  (class class class)co 6081  crio 6542  cgr 21774  GIdcgi 21775  cgn 21776 This theorem is referenced by:  ginvsn  21937  nvinvfval  22121 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781
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