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Theorem isexid2 21905
 Description: If , then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
isexid2.1
Assertion
Ref Expression
isexid2
Distinct variable groups:   ,,   ,,

Proof of Theorem isexid2
StepHypRef Expression
1 isexid2.1 . 2
2 rngopid 21903 . . . . 5
3 elin 3522 . . . . . . 7
4 eqid 2435 . . . . . . . . . . 11
54isexid 21897 . . . . . . . . . 10
65ibi 233 . . . . . . . . 9
76a1d 23 . . . . . . . 8
87adantl 453 . . . . . . 7
93, 8sylbi 188 . . . . . 6
10 eqeq2 2444 . . . . . . 7
11 raleq 2896 . . . . . . . 8
1211rexeqbi1dv 2905 . . . . . . 7
1310, 12imbi12d 312 . . . . . 6
149, 13syl5ibr 213 . . . . 5
152, 14mpcom 34 . . . 4
1615com12 29 . . 3
17 raleq 2896 . . . 4
1817rexeqbi1dv 2905 . . 3
1916, 18sylibrd 226 . 2
201, 19ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698   cin 3311   cdm 4870   crn 4871  (class class class)co 6073   cexid 21894  cmagm 21898 This theorem is referenced by:  exidu1  21906 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-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-exid 21895  df-mgm 21899
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