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Theorem exidres 26553
 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 26552 . . 3
5 oveq1 6088 . . . . . . 7
65eqeq1d 2444 . . . . . 6
7 oveq2 6089 . . . . . . 7
87eqeq1d 2444 . . . . . 6
96, 8anbi12d 692 . . . . 5
109ralbidv 2725 . . . 4
1110rspcev 3052 . . 3
124, 11syl 16 . 2
13 resexg 5185 . . . . 5
143, 13syl5eqel 2520 . . . 4
15 eqid 2436 . . . . 5
1615isexid 21905 . . . 4
1714, 16syl 16 . . 3
18173ad2ant1 978 . 2
1912, 18mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706  cvv 2956   cin 3319   wss 3320   cxp 4876   cdm 4878   crn 4879   cres 4880  cfv 5454  (class class class)co 6081  GIdcgi 21775   cexid 21902  cmagm 21906 This theorem is referenced by:  exidresid  26554 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-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-rmo 2713  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-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-ov 6084  df-riota 6549  df-gid 21780  df-exid 21903  df-mgm 21907
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