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Theorem fo1stres 6159
Description: Onto mapping of a restriction of the  1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
fo1stres  |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> A )

Proof of Theorem fo1stres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3477 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 opelxp 4735 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5558 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 1st `  <. x ,  y
>. ) )
4 vex 2804 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 2804 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op1st 6144 . . . . . . . . . . . 12  |-  ( 1st `  <. x ,  y
>. )  =  x
73, 6syl6req 2345 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f1stres 6157 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
9 ffn 5405 . . . . . . . . . . . . 13  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ( 1st  |`  ( A  X.  B ) )  Fn  ( A  X.  B
) )
108, 9ax-mp 8 . . . . . . . . . . . 12  |-  ( 1st  |`  ( A  X.  B
) )  Fn  ( A  X.  B )
11 fnfvelrn 5678 . . . . . . . . . . . 12  |-  ( ( ( 1st  |`  ( A  X.  B ) )  Fn  ( A  X.  B )  /\  <. x ,  y >.  e.  ( A  X.  B ) )  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1210, 11mpan 651 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2370 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
142, 13sylbir 204 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1514expcom 424 . . . . . . . 8  |-  ( y  e.  B  ->  (
x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1615exlimiv 1624 . . . . . . 7  |-  ( E. y  y  e.  B  ->  ( x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
171, 16sylbi 187 . . . . . 6  |-  ( B  =/=  (/)  ->  ( x  e.  A  ->  x  e. 
ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3198 . . . . 5  |-  ( B  =/=  (/)  ->  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) )
19 frn 5411 . . . . . 6  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ran  ( 1st  |`  ( A  X.  B ) )  C_  A )
208, 19ax-mp 8 . . . . 5  |-  ran  ( 1st  |`  ( A  X.  B ) )  C_  A
2118, 20jctil 523 . . . 4  |-  ( B  =/=  (/)  ->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3207 . . . 4  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 203 . . 3  |-  ( B  =/=  (/)  ->  ran  ( 1st  |`  ( A  X.  B
) )  =  A )
2423, 8jctil 523 . 2  |-  ( B  =/=  (/)  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5471 . 2  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) -onto-> A  <->  ( ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
2624, 25sylibr 203 1  |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   <.cop 3656    X. cxp 4703   ran crn 4706    |` cres 4707    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  1stconst  6223  txcmpb  17354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138
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