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Theorem exidres 31602
 Description: The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
Hypotheses
Ref Expression
exidres.1
exidres.2 GId
exidres.3
Assertion
Ref Expression
exidres

Proof of Theorem exidres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exidres.1 . . . 4
2 exidres.2 . . . 4 GId
3 exidres.3 . . . 4
41, 2, 3exidreslem 31601 . . 3
5 oveq1 6284 . . . . . . 7
65eqeq1d 2404 . . . . . 6
7 oveq2 6285 . . . . . . 7
87eqeq1d 2404 . . . . . 6
96, 8anbi12d 709 . . . . 5
109ralbidv 2842 . . . 4
1110rspcev 3159 . . 3
124, 11syl 17 . 2
13 resexg 5135 . . . . 5
143, 13syl5eqel 2494 . . . 4
15 eqid 2402 . . . . 5
1615isexid 25719 . . . 4
1714, 16syl 17 . . 3
18173ad2ant1 1018 . 2
1912, 18mpbird 232 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2753  wrex 2754  cvv 3058   cin 3412   wss 3413   cxp 4820   cdm 4822   crn 4823   cres 4824  cfv 5568  (class class class)co 6277  GIdcgi 25589   cexid 25716  cmagm 25720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-riota 6239  df-ov 6280  df-gid 25594  df-exid 25717  df-mgmOLD 25721 This theorem is referenced by:  exidresid  31603
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