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Theorem fo1stresm 6140
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 2233 . . 3  |-  ( v  =  y  ->  (
v  e.  B  <->  y  e.  B ) )
21cbvexv 1911 . 2  |-  ( E. v  v  e.  B  <->  E. y  y  e.  B
)
3 opelxp 4641 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  <->  ( u  e.  A  /\  v  e.  B ) )
4 fvres 5520 . . . . . . . . . . . 12  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  =  ( 1st `  <. u ,  v
>. ) )
5 vex 2733 . . . . . . . . . . . . 13  |-  u  e. 
_V
6 vex 2733 . . . . . . . . . . . . 13  |-  v  e. 
_V
75, 6op1st 6125 . . . . . . . . . . . 12  |-  ( 1st `  <. u ,  v
>. )  =  u
84, 7eqtr2di 2220 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. u ,  v
>. ) )
9 f1stres 6138 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
10 ffn 5347 . . . . . . . . . . . . 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 5628 . . . . . . . . . . . 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 422 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
148, 13eqeltrd 2247 . . . . . . . . . 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 1591 . . . . . . 7  |-  ( E. v  v  e.  B  ->  ( u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3153 . . . . . 6  |-  ( E. v  v  e.  B  ->  A  C_  ran  ( 1st  |`  ( A  X.  B
) ) )
19 frn 5356 . . . . . . 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 310 . . . . 5  |-  ( E. v  v  e.  B  ->  ( ran  ( 1st  |`  ( A  X.  B
) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3162 . . . . 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 310 . . 3  |-  ( E. v  v  e.  B  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5424 . . 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 1348   E.wex 1485    e. wcel 2141    C_ wss 3121   <.cop 3586    X. cxp 4609   ran crn 4612    |` cres 4613    Fn wfn 5193   -->wf 5194   -onto->wfo 5196   ` cfv 5198   1stc1st 6117
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119
This theorem is referenced by:  1stconst  6200
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