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Theorem foov 6010
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 5691 . 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 5541 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5877 . . . . . . 7  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2346 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eqeq2d 2307 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
65rexxp 4844 . . . 4  |-  ( E. w  e.  ( A  X.  B ) z  =  ( F `  w )  <->  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
76ralbii 2580 . . 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 1632   A.wral 2556   E.wrex 2557   <.cop 3656    X. cxp 4703   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874
This theorem is referenced by:  iunfictbso  7757  xpsff1o  13486  mndfo  14413  gafo  14766  isgrpo  20879  isgrpoi  20881  isgrp2d  20918  isgrpda  20980  opidon  21005  rngmgmbs4  21100  mgmlion  25440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877
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