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Theorem fo2ndres 6303
Description: Onto mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
fo2ndres  |-  ( A  =/=  (/)  ->  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> B )

Proof of Theorem fo2ndres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3573 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 opelxp 4841 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5678 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
4 vex 2895 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 2895 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op2nd 6288 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
73, 6syl6req 2429 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  =  ( ( 2nd  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f2ndres 6301 . . . . . . . . . . . . 13  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
9 ffn 5524 . . . . . . . . . . . . 13  |-  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  ->  ( 2nd  |`  ( A  X.  B ) )  Fn  ( A  X.  B
) )
108, 9ax-mp 8 . . . . . . . . . . . 12  |-  ( 2nd  |`  ( A  X.  B
) )  Fn  ( A  X.  B )
11 fnfvelrn 5799 . . . . . . . . . . . 12  |-  ( ( ( 2nd  |`  ( A  X.  B ) )  Fn  ( A  X.  B )  /\  <. x ,  y >.  e.  ( A  X.  B ) )  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
1210, 11mpan 652 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2454 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
142, 13sylbir 205 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
1514ex 424 . . . . . . . 8  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
1615exlimiv 1641 . . . . . . 7  |-  ( E. x  x  e.  A  ->  ( y  e.  B  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
171, 16sylbi 188 . . . . . 6  |-  ( A  =/=  (/)  ->  ( y  e.  B  ->  y  e. 
ran  ( 2nd  |`  ( A  X.  B ) ) ) )
1817ssrdv 3290 . . . . 5  |-  ( A  =/=  (/)  ->  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) )
19 frn 5530 . . . . . 6  |-  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  ->  ran  ( 2nd  |`  ( A  X.  B ) )  C_  B )
208, 19ax-mp 8 . . . . 5  |-  ran  ( 2nd  |`  ( A  X.  B ) )  C_  B
2118, 20jctil 524 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  ( 2nd  |`  ( A  X.  B ) )  C_  B  /\  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
22 eqss 3299 . . . 4  |-  ( ran  ( 2nd  |`  ( A  X.  B ) )  =  B  <->  ( ran  ( 2nd  |`  ( A  X.  B ) )  C_  B  /\  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 204 . . 3  |-  ( A  =/=  (/)  ->  ran  ( 2nd  |`  ( A  X.  B
) )  =  B )
2423, 8jctil 524 . 2  |-  ( A  =/=  (/)  ->  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  /\  ran  ( 2nd  |`  ( A  X.  B ) )  =  B ) )
25 dffo2 5590 . 2  |-  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) -onto-> B  <->  ( ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B  /\  ran  ( 2nd  |`  ( A  X.  B ) )  =  B ) )
2624, 25sylibr 204 1  |-  ( A  =/=  (/)  ->  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2543    C_ wss 3256   (/)c0 3564   <.cop 3753    X. cxp 4809   ran crn 4812    |` cres 4813    Fn wfn 5382   -->wf 5383   -onto->wfo 5385   ` cfv 5387   2ndc2nd 6280
This theorem is referenced by:  2ndconst  6368  txcmpb  17590
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 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fo 5393  df-fv 5395  df-2nd 6282
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