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Theorem fo2ndres 6824
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 3803 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 opelxp 5038 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5886 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
4 vex 3112 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 3112 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op2nd 6808 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
73, 6syl6req 2515 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  =  ( ( 2nd  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f2ndres 6822 . . . . . . . . . . . . 13  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
9 ffn 5737 . . . . . . . . . . . . 13  |-  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  ->  ( 2nd  |`  ( A  X.  B ) )  Fn  ( A  X.  B
) )
108, 9ax-mp 5 . . . . . . . . . . . 12  |-  ( 2nd  |`  ( A  X.  B
) )  Fn  ( A  X.  B )
11 fnfvelrn 6029 . . . . . . . . . . . 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 670 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2545 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
142, 13sylbir 213 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
1514ex 434 . . . . . . . 8  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
1615exlimiv 1723 . . . . . . 7  |-  ( E. x  x  e.  A  ->  ( y  e.  B  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
171, 16sylbi 195 . . . . . 6  |-  ( A  =/=  (/)  ->  ( y  e.  B  ->  y  e. 
ran  ( 2nd  |`  ( A  X.  B ) ) ) )
1817ssrdv 3505 . . . . 5  |-  ( A  =/=  (/)  ->  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) )
19 frn 5743 . . . . . 6  |-  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  ->  ran  ( 2nd  |`  ( A  X.  B ) )  C_  B )
208, 19ax-mp 5 . . . . 5  |-  ran  ( 2nd  |`  ( A  X.  B ) )  C_  B
2118, 20jctil 537 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  ( 2nd  |`  ( A  X.  B ) )  C_  B  /\  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
22 eqss 3514 . . . 4  |-  ( ran  ( 2nd  |`  ( A  X.  B ) )  =  B  <->  ( ran  ( 2nd  |`  ( A  X.  B ) )  C_  B  /\  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 212 . . 3  |-  ( A  =/=  (/)  ->  ran  ( 2nd  |`  ( A  X.  B
) )  =  B )
2423, 8jctil 537 . 2  |-  ( A  =/=  (/)  ->  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  /\  ran  ( 2nd  |`  ( A  X.  B ) )  =  B ) )
25 dffo2 5805 . 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 212 1  |-  ( A  =/=  (/)  ->  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652    C_ wss 3471   (/)c0 3793   <.cop 4038    X. cxp 5006   ran crn 5009    |` cres 5010    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-2nd 6800
This theorem is referenced by:  2ndconst  6888  txcmpb  20271
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