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Theorem foov 6237
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 5858 . 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 5691 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6094 . . . . . . 7  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2493 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eqeq2d 2454 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
65rexxp 4982 . . . 4  |-  ( E. w  e.  ( A  X.  B ) z  =  ( F `  w )  <->  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
76ralbii 2739 . . 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 1369   A.wral 2715   E.wrex 2716   <.cop 3883    X. cxp 4838   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-ov 6094
This theorem is referenced by:  iunfictbso  8284  xpsff1o  14506  mndfo  15445  gafo  15814  isgrpo  23683  isgrpoi  23685  isgrp2d  23722  isgrpda  23784  opidon  23809  rngmgmbs4  23904
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