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Theorem grpoidval 21794
 Description: Lemma for grpoidcl 21795 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpoidval.1
grpoidval.2 GId
Assertion
Ref Expression
grpoidval
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem grpoidval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 grpoidval.2 . 2 GId
2 grpoidval.1 . . . 4
32gidval 21791 . . 3 GId
4 simpl 444 . . . . . . . . 9
54ralimi 2773 . . . . . . . 8
65rgenw 2765 . . . . . . 7
76a1i 11 . . . . . 6
82grpoidinv 21786 . . . . . . 7
9 simpl 444 . . . . . . . . 9
109ralimi 2773 . . . . . . . 8
1110reximi 2805 . . . . . . 7
128, 11syl 16 . . . . . 6
132grpoideu 21787 . . . . . 6
147, 12, 133jca 1134 . . . . 5
15 reupick2 3619 . . . . 5
1614, 15sylan 458 . . . 4
1716riotabidva 6558 . . 3
183, 17eqtr4d 2470 . 2 GId
191, 18syl5eq 2479 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wrex 2698  wreu 2699   crn 4871  cfv 5446  (class class class)co 6073  crio 6534  cgr 21764  GIdcgi 21765 This theorem is referenced by:  grpoidcl  21795  grpoidinv2  21796  cnid  21929  mulid  21934  hilid  22653 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  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-fo 5452  df-fv 5454  df-ov 6076  df-riota 6541  df-grpo 21769  df-gid 21770
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