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Theorem rngopid 21301
Description: Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
rngopid  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )

Proof of Theorem rngopid
StepHypRef Expression
1 eqid 2366 . . 3  |-  dom  dom  G  =  dom  dom  G
21opidon 21300 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( dom  dom  G  X.  dom  dom  G ) -onto-> dom 
dom  G )
3 forn 5560 . 2  |-  ( G : ( dom  dom  G  X.  dom  dom  G
) -onto-> dom  dom  G  ->  ran 
G  =  dom  dom  G )
42, 3syl 15 1  |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran  G  =  dom  dom  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715    i^i cin 3237    X. cxp 4790   dom cdm 4792   ran crn 4793   -onto->wfo 5356    ExId cexid 21292   Magmacmagm 21296
This theorem is referenced by:  isexid2  21303  ismndo2  21323  exidcl  26072  exidresid  26075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-ov 5984  df-exid 21293  df-mgm 21297
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