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Theorem fo1stres 6797
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 3793 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 opelxp 5018 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5862 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 1st `  <. x ,  y
>. ) )
4 vex 3109 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 3109 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op1st 6781 . . . . . . . . . . . 12  |-  ( 1st `  <. x ,  y
>. )  =  x
73, 6syl6req 2512 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f1stres 6795 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
9 ffn 5713 . . . . . . . . . . . . 13  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ( 1st  |`  ( A  X.  B ) )  Fn  ( A  X.  B
) )
108, 9ax-mp 5 . . . . . . . . . . . 12  |-  ( 1st  |`  ( A  X.  B
) )  Fn  ( A  X.  B )
11 fnfvelrn 6004 . . . . . . . . . . . 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 668 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2542 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
142, 13sylbir 213 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1514expcom 433 . . . . . . . 8  |-  ( y  e.  B  ->  (
x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1615exlimiv 1727 . . . . . . 7  |-  ( E. y  y  e.  B  ->  ( x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
171, 16sylbi 195 . . . . . 6  |-  ( B  =/=  (/)  ->  ( x  e.  A  ->  x  e. 
ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3495 . . . . 5  |-  ( B  =/=  (/)  ->  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) )
19 frn 5719 . . . . . 6  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ran  ( 1st  |`  ( A  X.  B ) )  C_  A )
208, 19ax-mp 5 . . . . 5  |-  ran  ( 1st  |`  ( A  X.  B ) )  C_  A
2118, 20jctil 535 . . . 4  |-  ( B  =/=  (/)  ->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3504 . . . 4  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 212 . . 3  |-  ( B  =/=  (/)  ->  ran  ( 1st  |`  ( A  X.  B
) )  =  A )
2423, 8jctil 535 . 2  |-  ( B  =/=  (/)  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5781 . 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 212 1  |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783   <.cop 4022    X. cxp 4986   ran crn 4989    |` cres 4990    Fn wfn 5565   -->wf 5566   -onto->wfo 5568   ` cfv 5570   1stc1st 6771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773
This theorem is referenced by:  1stconst  6861  txcmpb  20311
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