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Theorem fo1stresm 6025
Description: Onto mapping of a restriction of the  1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
Assertion
Ref Expression
fo1stresm  |-  ( E. y  y  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
Distinct variable group:    y, B
Allowed substitution hint:    A( y)

Proof of Theorem fo1stresm
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2178 . . 3  |-  ( v  =  y  ->  (
v  e.  B  <->  y  e.  B ) )
21cbvexv 1870 . 2  |-  ( E. v  v  e.  B  <->  E. y  y  e.  B
)
3 opelxp 4537 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  <->  ( u  e.  A  /\  v  e.  B ) )
4 fvres 5411 . . . . . . . . . . . 12  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  =  ( 1st `  <. u ,  v
>. ) )
5 vex 2661 . . . . . . . . . . . . 13  |-  u  e. 
_V
6 vex 2661 . . . . . . . . . . . . 13  |-  v  e. 
_V
75, 6op1st 6010 . . . . . . . . . . . 12  |-  ( 1st `  <. u ,  v
>. )  =  u
84, 7syl6req 2165 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. u ,  v
>. ) )
9 f1stres 6023 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
10 ffn 5240 . . . . . . . . . . . . 13  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ( 1st  |`  ( A  X.  B ) )  Fn  ( A  X.  B
) )
119, 10ax-mp 5 . . . . . . . . . . . 12  |-  ( 1st  |`  ( A  X.  B
) )  Fn  ( A  X.  B )
12 fnfvelrn 5518 . . . . . . . . . . . 12  |-  ( ( ( 1st  |`  ( A  X.  B ) )  Fn  ( A  X.  B )  /\  <. u ,  v >.  e.  ( A  X.  B ) )  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1311, 12mpan 418 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
148, 13eqeltrd 2192 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
153, 14sylbir 134 . . . . . . . . 9  |-  ( ( u  e.  A  /\  v  e.  B )  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1615expcom 115 . . . . . . . 8  |-  ( v  e.  B  ->  (
u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1716exlimiv 1560 . . . . . . 7  |-  ( E. v  v  e.  B  ->  ( u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3071 . . . . . 6  |-  ( E. v  v  e.  B  ->  A  C_  ran  ( 1st  |`  ( A  X.  B
) ) )
19 frn 5249 . . . . . . 7  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ran  ( 1st  |`  ( A  X.  B ) )  C_  A )
209, 19ax-mp 5 . . . . . 6  |-  ran  ( 1st  |`  ( A  X.  B ) )  C_  A
2118, 20jctil 308 . . . . 5  |-  ( E. v  v  e.  B  ->  ( ran  ( 1st  |`  ( A  X.  B
) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3080 . . . . 5  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 133 . . . 4  |-  ( E. v  v  e.  B  ->  ran  ( 1st  |`  ( A  X.  B ) )  =  A )
2423, 9jctil 308 . . 3  |-  ( E. v  v  e.  B  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5317 . . 3  |-  ( ( 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 133 . 2  |-  ( E. v  v  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
272, 26sylbir 134 1  |-  ( E. y  y  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463    C_ wss 3039   <.cop 3498    X. cxp 4505   ran crn 4508    |` cres 4509    Fn wfn 5086   -->wf 5087   -onto->wfo 5089   ` cfv 5091   1stc1st 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-1st 6004
This theorem is referenced by:  1stconst  6084
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