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Theorem fo2ndres 6363
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 3629 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 opelxp 4900 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5737 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
4 vex 2951 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 2951 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op2nd 6348 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
73, 6syl6req 2484 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  =  ( ( 2nd  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f2ndres 6361 . . . . . . . . . . . . 13  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
9 ffn 5583 . . . . . . . . . . . . 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 5859 . . . . . . . . . . . 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 2509 . . . . . . . . . 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 1644 . . . . . . 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 3346 . . . . 5  |-  ( A  =/=  (/)  ->  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) )
19 frn 5589 . . . . . 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 3355 . . . 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 5649 . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   <.cop 3809    X. cxp 4868   ran crn 4871    |` cres 4872    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   ` cfv 5446   2ndc2nd 6340
This theorem is referenced by:  2ndconst  6428  txcmpb  17668
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  ax-un 4693
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-csb 3244  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-iun 4087  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  df-2nd 6342
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