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

Theorem foco2 5878
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 988 . 2  |-  ( ( F : B --> C  /\  G : A --> B  /\  ( F  o.  G
) : A -onto-> C
)  ->  F : B
--> C )
2 foelrn 5877 . . . . . 6  |-  ( ( ( F  o.  G
) : A -onto-> C  /\  y  e.  C
)  ->  E. z  e.  A  y  =  ( ( F  o.  G ) `  z
) )
3 ffvelrn 5856 . . . . . . . . . 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 5783 . . . . . . . . . 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 5706 . . . . . . . . . . 11  |-  ( x  =  ( G `  z )  ->  ( F `  x )  =  ( F `  ( G `  z ) ) )
87eqeq2d 2454 . . . . . . . . . 10  |-  ( x  =  ( G `  z )  ->  (
( ( F  o.  G ) `  z
)  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  ( G `
 z ) ) ) )
98rspcev 3088 . . . . . . . . 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 2449 . . . . . . . . 9  |-  ( y  =  ( ( F  o.  G ) `  z )  ->  (
y  =  ( F `
 x )  <->  ( ( F  o.  G ) `  z )  =  ( F `  x ) ) )
1211rexbidv 2751 . . . . . . . 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 2856 . . . . . 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 2814 . . 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 1182 . 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 5873 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2730   E.wrex 2731    o. ccom 4859   -->wf 5429   -onto->wfo 5431   ` cfv 5433
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546
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 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-fo 5439  df-fv 5441
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator