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Theorem foco2 5829
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 957 . 2  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B
--> C )
2 foelrn 5828 . . . . . 6  |-  ( ( ( F  o.  G
) : A -onto-> C  /\  y  e.  C
)  ->  E. z  e.  A  y  =  ( ( F  o.  G ) `  z
) )
3 ffvelrn 5808 . . . . . . . . . 10  |-  ( ( G : A --> B  /\  z  e.  A )  ->  ( G `  z
)  e.  B )
43adantll 695 . . . . . . . . 9  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  ( G `  z )  e.  B )
5 fvco3 5740 . . . . . . . . . 10  |-  ( ( G : A --> B  /\  z  e.  A )  ->  ( ( F  o.  G ) `  z
)  =  ( F `
 ( G `  z ) ) )
65adantll 695 . . . . . . . . 9  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  (
( F  o.  G
) `  z )  =  ( F `  ( G `  z ) ) )
7 fveq2 5669 . . . . . . . . . . 11  |-  ( x  =  ( G `  z )  ->  ( F `  x )  =  ( F `  ( G `  z ) ) )
87eqeq2d 2399 . . . . . . . . . 10  |-  ( x  =  ( G `  z )  ->  (
( ( F  o.  G ) `  z
)  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  ( G `
 z ) ) ) )
98rspcev 2996 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  E. x  e.  B  ( ( F  o.  G ) `  z )  =  ( F `  x ) )
11 eqeq1 2394 . . . . . . . . 9  |-  ( y  =  ( ( F  o.  G ) `  z )  ->  (
y  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  x ) ) )
1211rexbidv 2671 . . . . . . . 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 214 . . . . . . 7  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  (
y  =  ( ( F  o.  G ) `
 z )  ->  E. x  e.  B  y  =  ( F `  x ) ) )
1413rexlimdva 2774 . . . . . 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 30 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( ( ( F  o.  G ) : A -onto-> C  /\  y  e.  C )  ->  E. x  e.  B  y  =  ( F `  x ) ) )
1615impl 604 . . . 4  |-  ( ( ( ( F : B
--> C  /\  G : A
--> B )  /\  ( F  o.  G ) : A -onto-> C )  /\  y  e.  C )  ->  E. x  e.  B  y  =  ( F `  x ) )
1716ralrimiva 2733 . . 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 1148 . 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 5824 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. y  e.  C  E. x  e.  B  y  =  ( F `  x ) ) )
201, 18, 19sylanbrc 646 1  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B -onto-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   E.wrex 2651    o. ccom 4823   -->wf 5391   -onto->wfo 5393   ` cfv 5395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403
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