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Theorem fo1stres 6700
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 3744 . . . . . . 7  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
2 opelxp 4967 . . . . . . . . . 10  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 fvres 5803 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  =  ( 1st `  <. x ,  y
>. ) )
4 vex 3071 . . . . . . . . . . . . 13  |-  x  e. 
_V
5 vex 3071 . . . . . . . . . . . . 13  |-  y  e. 
_V
64, 5op1st 6685 . . . . . . . . . . . 12  |-  ( 1st `  <. x ,  y
>. )  =  x
73, 6syl6req 2509 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  x  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. x ,  y
>. ) )
8 f1stres 6698 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
9 ffn 5657 . . . . . . . . . . . . 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 5939 . . . . . . . . . . . 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 670 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. x ,  y >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
137, 12eqeltrd 2539 . . . . . . . . . 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 435 . . . . . . . 8  |-  ( y  e.  B  ->  (
x  e.  A  ->  x  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1615exlimiv 1689 . . . . . . 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 3460 . . . . 5  |-  ( B  =/=  (/)  ->  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) )
19 frn 5663 . . . . . 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 537 . . . 4  |-  ( B  =/=  (/)  ->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3469 . . . 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 537 . 2  |-  ( B  =/=  (/)  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5722 . 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 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644    C_ wss 3426   (/)c0 3735   <.cop 3981    X. cxp 4936   ran crn 4939    |` cres 4940    Fn wfn 5511   -->wf 5512   -onto->wfo 5514   ` cfv 5516   1stc1st 6675
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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fo 5522  df-fv 5524  df-1st 6677
This theorem is referenced by:  1stconst  6761  txcmpb  19333
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