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Theorem foov 6444
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 6047 . 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 5872 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6298 . . . . . . 7  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2526 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eqeq2d 2481 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
65rexxp 5151 . . . 4  |-  ( E. w  e.  ( A  X.  B ) z  =  ( F `  w )  <->  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
76ralbii 2898 . . 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 694 . 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 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   A.wral 2817   E.wrex 2818   <.cop 4039    X. cxp 5003   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-ov 6298
This theorem is referenced by:  iunfictbso  8507  xpsff1o  14840  mndpfo  15817  gafo  16206  isgrpo  25021  isgrpoi  25023  isgrp2d  25060  isgrpda  25122  opidonOLD  25147  rngmgmbs4  25242
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