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Theorem fo1stres 6310
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 3581 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 opelxp 4849 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5686 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 1st `  <. x ,  y
>. ) )
4 vex 2903 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 2903 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op1st 6295 . . . . . . . . . . . 12  |-  ( 1st `  <. x ,  y
>. )  =  x
73, 6syl6req 2437 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f1stres 6308 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
9 ffn 5532 . . . . . . . . . . . . 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 5807 . . . . . . . . . . . 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 652 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2462 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
142, 13sylbir 205 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1514expcom 425 . . . . . . . 8  |-  ( y  e.  B  ->  (
x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1615exlimiv 1641 . . . . . . 7  |-  ( E. y  y  e.  B  ->  ( x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
171, 16sylbi 188 . . . . . 6  |-  ( B  =/=  (/)  ->  ( x  e.  A  ->  x  e. 
ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3298 . . . . 5  |-  ( B  =/=  (/)  ->  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) )
19 frn 5538 . . . . . 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 524 . . . 4  |-  ( B  =/=  (/)  ->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3307 . . . 4  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 204 . . 3  |-  ( B  =/=  (/)  ->  ran  ( 1st  |`  ( A  X.  B
) )  =  A )
2423, 8jctil 524 . 2  |-  ( B  =/=  (/)  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5598 . 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 204 1  |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551    C_ wss 3264   (/)c0 3572   <.cop 3761    X. cxp 4817   ran crn 4820    |` cres 4821    Fn wfn 5390   -->wf 5391   -onto->wfo 5393   ` cfv 5395   1stc1st 6287
This theorem is referenced by:  1stconst  6375  txcmpb  17598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-1st 6289
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