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Theorem fo1stres 6817
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 3741 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 opelxp 4864 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5879 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 1st `  <. x ,  y
>. ) )
4 vex 3048 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 3048 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op1st 6801 . . . . . . . . . . . 12  |-  ( 1st `  <. x ,  y
>. )  =  x
73, 6syl6req 2502 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f1stres 6815 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
9 ffn 5728 . . . . . . . . . . . . 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 6019 . . . . . . . . . . . 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 676 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2529 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
142, 13sylbir 217 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1514expcom 437 . . . . . . . 8  |-  ( y  e.  B  ->  (
x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1615exlimiv 1776 . . . . . . 7  |-  ( E. y  y  e.  B  ->  ( x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
171, 16sylbi 199 . . . . . 6  |-  ( B  =/=  (/)  ->  ( x  e.  A  ->  x  e. 
ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3438 . . . . 5  |-  ( B  =/=  (/)  ->  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) )
19 frn 5735 . . . . . 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 540 . . . 4  |-  ( B  =/=  (/)  ->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3447 . . . 4  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 216 . . 3  |-  ( B  =/=  (/)  ->  ran  ( 1st  |`  ( A  X.  B
) )  =  A )
2423, 8jctil 540 . 2  |-  ( B  =/=  (/)  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5797 . 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 216 1  |-  ( B  =/=  (/)  ->  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B )
-onto-> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622    C_ wss 3404   (/)c0 3731   <.cop 3974    X. cxp 4832   ran crn 4835    |` cres 4836    Fn wfn 5577   -->wf 5578   -onto->wfo 5580   ` cfv 5582   1stc1st 6791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-1st 6793
This theorem is referenced by:  1stconst  6884  txcmpb  20659
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