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Theorem foco2 5763
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 955 . 2  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B
--> C )
2 foelrn 5762 . . . . . 6  |-  ( ( ( F  o.  G
) : A -onto-> C  /\  y  e.  C
)  ->  E. z  e.  A  y  =  ( ( F  o.  G ) `  z
) )
3 ffvelrn 5746 . . . . . . . . . 10  |-  ( ( G : A --> B  /\  z  e.  A )  ->  ( G `  z
)  e.  B )
43adantll 694 . . . . . . . . 9  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  ( G `  z )  e.  B )
5 fvco3 5679 . . . . . . . . . 10  |-  ( ( G : A --> B  /\  z  e.  A )  ->  ( ( F  o.  G ) `  z
)  =  ( F `
 ( G `  z ) ) )
65adantll 694 . . . . . . . . 9  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  (
( F  o.  G
) `  z )  =  ( F `  ( G `  z ) ) )
7 fveq2 5608 . . . . . . . . . . 11  |-  ( x  =  ( G `  z )  ->  ( F `  x )  =  ( F `  ( G `  z ) ) )
87eqeq2d 2369 . . . . . . . . . 10  |-  ( x  =  ( G `  z )  ->  (
( ( F  o.  G ) `  z
)  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  ( G `
 z ) ) ) )
98rspcev 2960 . . . . . . . . 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 642 . . . . . . . 8  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  E. x  e.  B  ( ( F  o.  G ) `  z )  =  ( F `  x ) )
11 eqeq1 2364 . . . . . . . . 9  |-  ( y  =  ( ( F  o.  G ) `  z )  ->  (
y  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  x ) ) )
1211rexbidv 2640 . . . . . . . 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 213 . . . . . . 7  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  z  e.  A )  ->  (
y  =  ( ( F  o.  G ) `
 z )  ->  E. x  e.  B  y  =  ( F `  x ) ) )
1413rexlimdva 2743 . . . . . 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 28 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( ( ( F  o.  G ) : A -onto-> C  /\  y  e.  C )  ->  E. x  e.  B  y  =  ( F `  x ) ) )
1615impl 603 . . . 4  |-  ( ( ( ( F : B
--> C  /\  G : A
--> B )  /\  ( F  o.  G ) : A -onto-> C )  /\  y  e.  C )  ->  E. x  e.  B  y  =  ( F `  x ) )
1716ralrimiva 2702 . . 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 1146 . 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 5758 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. y  e.  C  E. x  e.  B  y  =  ( F `  x ) ) )
201, 18, 19sylanbrc 645 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620    o. ccom 4775   -->wf 5333   -onto->wfo 5335   ` cfv 5337
This theorem is referenced by:  foco2OLD  25695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345
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