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Theorem fo2ndres 6714
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 3757 . . . . . . 7  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 opelxp 4980 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5816 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 2nd  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 2nd `  <. x ,  y
>. ) )
4 vex 3081 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 3081 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op2nd 6699 . . . . . . . . . . . 12  |-  ( 2nd `  <. x ,  y
>. )  =  y
73, 6syl6req 2512 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  y  =  ( ( 2nd  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f2ndres 6712 . . . . . . . . . . . . 13  |-  ( 2nd  |`  ( A  X.  B
) ) : ( A  X.  B ) --> B
9 ffn 5670 . . . . . . . . . . . . 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 5952 . . . . . . . . . . . 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 2542 . . . . . . . . . 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 1689 . . . . . . 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 3473 . . . . 5  |-  ( A  =/=  (/)  ->  B  C_  ran  ( 2nd  |`  ( A  X.  B ) ) )
19 frn 5676 . . . . . 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 3482 . . . 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 5735 . 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 1370   E.wex 1587    e. wcel 1758    =/= wne 2648    C_ wss 3439   (/)c0 3748   <.cop 3994    X. cxp 4949   ran crn 4952    |` cres 4953    Fn wfn 5524   -->wf 5525   -onto->wfo 5527   ` cfv 5529   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-2nd 6691
This theorem is referenced by:  2ndconst  6775  txcmpb  19359
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