MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foco2 Structured version   Unicode version

Theorem foco2 5881
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 5880 . . . . . 6  |-  ( ( ( F  o.  G
) : A -onto-> C  /\  y  e.  C
)  ->  E. z  e.  A  y  =  ( ( F  o.  G ) `  z
) )
3 ffvelrn 5860 . . . . . . . . . 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 5792 . . . . . . . . . 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 5720 . . . . . . . . . . 11  |-  ( x  =  ( G `  z )  ->  ( F `  x )  =  ( F `  ( G `  z ) ) )
87eqeq2d 2446 . . . . . . . . . 10  |-  ( x  =  ( G `  z )  ->  (
( ( F  o.  G ) `  z
)  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  ( G `
 z ) ) ) )
98rspcev 3044 . . . . . . . . 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 2441 . . . . . . . . 9  |-  ( y  =  ( ( F  o.  G ) `  z )  ->  (
y  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  x ) ) )
1211rexbidv 2718 . . . . . . . 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 2822 . . . . . 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 2781 . . 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 5876 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    o. ccom 4874   -->wf 5442   -onto->wfo 5444   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454
  Copyright terms: Public domain W3C validator