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Theorem fo2ndres 6144
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 3464 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 opelxp 4719 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5542 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
4 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op2nd 6129 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
73, 6syl6req 2332 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  =  ( ( 2nd  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f2ndres 6142 . . . . . . . . . . . . 13  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
9 ffn 5389 . . . . . . . . . . . . 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 5662 . . . . . . . . . . . 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 651 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2357 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
142, 13sylbir 204 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) )
1514ex 423 . . . . . . . 8  |-  ( x  e.  A  ->  (
y  e.  B  -> 
y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
1615exlimiv 1666 . . . . . . 7  |-  ( E. x  x  e.  A  ->  ( y  e.  B  ->  y  e.  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
171, 16sylbi 187 . . . . . 6  |-  ( A  =/=  (/)  ->  ( y  e.  B  ->  y  e. 
ran  ( 2nd  |`  ( A  X.  B ) ) ) )
1817ssrdv 3185 . . . . 5  |-  ( A  =/=  (/)  ->  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) )
19 frn 5395 . . . . . 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 523 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  ( 2nd  |`  ( A  X.  B ) )  C_  B  /\  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
22 eqss 3194 . . . 4  |-  ( ran  ( 2nd  |`  ( A  X.  B ) )  =  B  <->  ( ran  ( 2nd  |`  ( A  X.  B ) )  C_  B  /\  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 203 . . 3  |-  ( A  =/=  (/)  ->  ran  ( 2nd  |`  ( A  X.  B
) )  =  B )
2423, 8jctil 523 . 2  |-  ( A  =/=  (/)  ->  ( ( 2nd  |`  ( A  X.  B ) ) : ( A  X.  B
) --> B  /\  ran  ( 2nd  |`  ( A  X.  B ) )  =  B ) )
25 dffo2 5455 . 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 203 1  |-  ( A  =/=  (/)  ->  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   <.cop 3643    X. cxp 4687   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255   2ndc2nd 6121
This theorem is referenced by:  2ndconst  6208  txcmpb  17338  domrancur1b  25200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-2nd 6123
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