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Theorem grpofo 21792
 Description: A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1
Assertion
Ref Expression
grpofo

Proof of Theorem grpofo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpfo.1 . . . . . 6
21isgrpo 21789 . . . . 5
32ibi 234 . . . 4
43simp1d 970 . . 3
51eqcomi 2442 . . 3
64, 5jctir 526 . 2
7 dffo2 5660 . 2
86, 7sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  wrex 2708   cxp 4879   crn 4882  wf 5453  wfo 5455  (class class class)co 6084  cgr 21779 This theorem is referenced by:  grpocl  21793  grporndm  21803  grporn  21805  resgrprn  21873  subgores  21897  issubgoi  21903  rngosn  21997  rngodm1dm2  22011  rngosn3  22019  vcoprnelem  22062  nvgf  22102  ghomfo  25107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-ov 6087  df-grpo 21784
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