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Theorem opidon 21903
 Description: An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
opidon.1
Assertion
Ref Expression
opidon

Proof of Theorem opidon
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3554 . . . 4
21sseli 3337 . . 3
3 opidon.1 . . . . 5
43ismgm 21901 . . . 4
54ibi 233 . . 3
62, 5syl 16 . 2
7 inss2 3555 . . . . 5
87sseli 3337 . . . 4
93isexid 21898 . . . . 5
109biimpd 199 . . . 4
118, 8, 10sylc 58 . . 3
12 simpl 444 . . . . . . . 8
1312ralimi 2774 . . . . . . 7
14 oveq2 6082 . . . . . . . . . 10
15 id 20 . . . . . . . . . 10
1614, 15eqeq12d 2450 . . . . . . . . 9
1716rspcv 3041 . . . . . . . 8
18 eqcom 2438 . . . . . . . . . . 11
1914eqeq1d 2444 . . . . . . . . . . 11
2018, 19syl5bb 249 . . . . . . . . . 10
2120rspcev 3045 . . . . . . . . 9
2221ex 424 . . . . . . . 8
2317, 22syld 42 . . . . . . 7
2413, 23syl5 30 . . . . . 6
2524reximdv 2810 . . . . 5
2625impcom 420 . . . 4
2726ralrimiva 2782 . . 3
2811, 27syl 16 . 2
29 foov 6213 . 2
306, 28, 29sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2698  wrex 2699   cin 3312   cxp 4869   cdm 4871  wf 5443  wfo 5445  (class class class)co 6074   cexid 21895  cmagm 21899 This theorem is referenced by:  rngopid  21904  opidon2  21905 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 4323  ax-nul 4331  ax-pr 4396  ax-un 4694 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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-ov 6077  df-exid 21896  df-mgm 21900
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