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Theorem fo1stresm 5816
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 2116 . . 3  |-  ( v  =  y  ->  (
v  e.  B  <->  y  e.  B ) )
21cbvexv 1811 . 2  |-  ( E. v  v  e.  B  <->  E. y  y  e.  B
)
3 opelxp 4402 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  <->  ( u  e.  A  /\  v  e.  B ) )
4 fvres 5226 . . . . . . . . . . . 12  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  =  ( 1st `  <. u ,  v
>. ) )
5 vex 2577 . . . . . . . . . . . . 13  |-  u  e. 
_V
6 vex 2577 . . . . . . . . . . . . 13  |-  v  e. 
_V
75, 6op1st 5801 . . . . . . . . . . . 12  |-  ( 1st `  <. u ,  v
>. )  =  u
84, 7syl6req 2105 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. u ,  v
>. ) )
9 f1stres 5814 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
10 ffn 5074 . . . . . . . . . . . . 13  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ( 1st  |`  ( A  X.  B ) )  Fn  ( A  X.  B
) )
119, 10ax-mp 7 . . . . . . . . . . . 12  |-  ( 1st  |`  ( A  X.  B
) )  Fn  ( A  X.  B )
12 fnfvelrn 5327 . . . . . . . . . . . 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 408 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
148, 13eqeltrd 2130 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
153, 14sylbir 129 . . . . . . . . 9  |-  ( ( u  e.  A  /\  v  e.  B )  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1615expcom 113 . . . . . . . 8  |-  ( v  e.  B  ->  (
u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1716exlimiv 1505 . . . . . . 7  |-  ( E. v  v  e.  B  ->  ( u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 2979 . . . . . 6  |-  ( E. v  v  e.  B  ->  A  C_  ran  ( 1st  |`  ( A  X.  B
) ) )
19 frn 5080 . . . . . . 7  |-  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B
) --> A  ->  ran  ( 1st  |`  ( A  X.  B ) )  C_  A )
209, 19ax-mp 7 . . . . . 6  |-  ran  ( 1st  |`  ( A  X.  B ) )  C_  A
2118, 20jctil 299 . . . . 5  |-  ( E. v  v  e.  B  ->  ( ran  ( 1st  |`  ( A  X.  B
) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 2988 . . . . 5  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 141 . . . 4  |-  ( E. v  v  e.  B  ->  ran  ( 1st  |`  ( A  X.  B ) )  =  A )
2423, 9jctil 299 . . 3  |-  ( E. v  v  e.  B  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5138 . . 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 141 . 2  |-  ( E. v  v  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
272, 26sylbir 129 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 101    = wceq 1259   E.wex 1397    e. wcel 1409    C_ wss 2945   <.cop 3406    X. cxp 4371   ran crn 4374    |` cres 4375    Fn wfn 4925   -->wf 4926   -onto->wfo 4928   ` cfv 4930   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795
This theorem is referenced by:  1stconst  5870
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