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Theorem foco2 6052
Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
foco2  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B -onto-> C )

Proof of Theorem foco2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 996 . 2  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B
--> C )
2 foelrn 6051 . . . . . 6  |-  ( ( ( F  o.  G
) : A -onto-> C  /\  y  e.  C
)  ->  E. z  e.  A  y  =  ( ( F  o.  G ) `  z
) )
3 ffvelrn 6030 . . . . . . . . . 10  |-  ( ( G : A --> B  /\  z  e.  A )  ->  ( G `  z
)  e.  B )
43adantll 713 . . . . . . . . 9  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  ( G `  z )  e.  B )
5 fvco3 5951 . . . . . . . . . 10  |-  ( ( G : A --> B  /\  z  e.  A )  ->  ( ( F  o.  G ) `  z
)  =  ( F `
 ( G `  z ) ) )
65adantll 713 . . . . . . . . 9  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  (
( F  o.  G
) `  z )  =  ( F `  ( G `  z ) ) )
7 fveq2 5872 . . . . . . . . . . 11  |-  ( x  =  ( G `  z )  ->  ( F `  x )  =  ( F `  ( G `  z ) ) )
87eqeq2d 2481 . . . . . . . . . 10  |-  ( x  =  ( G `  z )  ->  (
( ( F  o.  G ) `  z
)  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  ( G `
 z ) ) ) )
98rspcev 3219 . . . . . . . . 9  |-  ( ( ( G `  z
)  e.  B  /\  ( ( F  o.  G ) `  z
)  =  ( F `
 ( G `  z ) ) )  ->  E. x  e.  B  ( ( F  o.  G ) `  z
)  =  ( F `
 x ) )
104, 6, 9syl2anc 661 . . . . . . . 8  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  E. x  e.  B  ( ( F  o.  G ) `  z )  =  ( F `  x ) )
11 eqeq1 2471 . . . . . . . . 9  |-  ( y  =  ( ( F  o.  G ) `  z )  ->  (
y  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  x ) ) )
1211rexbidv 2978 . . . . . . . 8  |-  ( y  =  ( ( F  o.  G ) `  z )  ->  ( E. x  e.  B  y  =  ( F `  x )  <->  E. x  e.  B  ( ( F  o.  G ) `  z )  =  ( F `  x ) ) )
1310, 12syl5ibrcom 222 . . . . . . 7  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  (
y  =  ( ( F  o.  G ) `
 z )  ->  E. x  e.  B  y  =  ( F `  x ) ) )
1413rexlimdva 2959 . . . . . 6  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( E. z  e.  A  y  =  ( ( F  o.  G ) `  z
)  ->  E. x  e.  B  y  =  ( F `  x ) ) )
152, 14syl5 32 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( ( ( F  o.  G ) : A -onto-> C  /\  y  e.  C )  ->  E. x  e.  B  y  =  ( F `  x ) ) )
1615impl 620 . . . 4  |-  ( ( ( ( F : B
--> C  /\  G : A
--> B )  /\  ( F  o.  G ) : A -onto-> C )  /\  y  e.  C )  ->  E. x  e.  B  y  =  ( F `  x ) )
1716ralrimiva 2881 . . 3  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  ( F  o.  G ) : A -onto-> C )  ->  A. y  e.  C  E. x  e.  B  y  =  ( F `  x ) )
18173impa 1191 . 2  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  A. y  e.  C  E. x  e.  B  y  =  ( F `  x ) )
19 dffo3 6047 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. y  e.  C  E. x  e.  B  y  =  ( F `  x ) ) )
201, 18, 19sylanbrc 664 1  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B -onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    o. ccom 5009   -->wf 5590   -onto->wfo 5592   ` cfv 5594
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-8 1769  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-pow 4631  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-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-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-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602
This theorem is referenced by: (None)
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