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Theorem foov 5994
Description: An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)
Assertion
Ref Expression
foov  |-  ( F : ( A  X.  B ) -onto-> C  <->  ( F : ( A  X.  B ) --> C  /\  A. z  e.  C  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    z, C    x, F, y, z
Allowed substitution hints:    C( x, y)

Proof of Theorem foov
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dffo3 5675 . 2  |-  ( F : ( A  X.  B ) -onto-> C  <->  ( F : ( A  X.  B ) --> C  /\  A. z  e.  C  E. w  e.  ( A  X.  B ) z  =  ( F `  w
) ) )
2 fveq2 5525 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5861 . . . . . . 7  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2333 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eqeq2d 2294 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
65rexxp 4828 . . . 4  |-  ( E. w  e.  ( A  X.  B ) z  =  ( F `  w )  <->  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
76ralbii 2567 . . 3  |-  ( A. z  e.  C  E. w  e.  ( A  X.  B ) z  =  ( F `  w
)  <->  A. z  e.  C  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
87anbi2i 675 . 2  |-  ( ( F : ( A  X.  B ) --> C  /\  A. z  e.  C  E. w  e.  ( A  X.  B
) z  =  ( F `  w ) )  <->  ( F :
( A  X.  B
) --> C  /\  A. z  e.  C  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) ) )
91, 8bitri 240 1  |-  ( F : ( A  X.  B ) -onto-> C  <->  ( F : ( A  X.  B ) --> C  /\  A. z  e.  C  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   E.wrex 2544   <.cop 3643    X. cxp 4687   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  iunfictbso  7741  xpsff1o  13470  mndfo  14397  gafo  14750  isgrpo  20863  isgrpoi  20865  isgrp2d  20902  isgrpda  20964  opidon  20989  rngmgmbs4  21084  mgmlion  25337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861
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