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Theorem fo1stresm 5932
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 2150 . . 3  |-  ( v  =  y  ->  (
v  e.  B  <->  y  e.  B ) )
21cbvexv 1843 . 2  |-  ( E. v  v  e.  B  <->  E. y  y  e.  B
)
3 opelxp 4467 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  <->  ( u  e.  A  /\  v  e.  B ) )
4 fvres 5329 . . . . . . . . . . . 12  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  =  ( 1st `  <. u ,  v
>. ) )
5 vex 2622 . . . . . . . . . . . . 13  |-  u  e. 
_V
6 vex 2622 . . . . . . . . . . . . 13  |-  v  e. 
_V
75, 6op1st 5917 . . . . . . . . . . . 12  |-  ( 1st `  <. u ,  v
>. )  =  u
84, 7syl6req 2137 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  =  ( ( 1st  |`  ( A  X.  B ) ) `
 <. u ,  v
>. ) )
9 f1stres 5930 . . . . . . . . . . . . 13  |-  ( 1st  |`  ( A  X.  B
) ) : ( A  X.  B ) --> A
10 ffn 5161 . . . . . . . . . . . . 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 5431 . . . . . . . . . . . 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 415 . . . . . . . . . . 11  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  ( ( 1st  |`  ( A  X.  B ) ) `  <. u ,  v >.
)  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
148, 13eqeltrd 2164 . . . . . . . . . 10  |-  ( <.
u ,  v >.  e.  ( A  X.  B
)  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
153, 14sylbir 133 . . . . . . . . 9  |-  ( ( u  e.  A  /\  v  e.  B )  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) )
1615expcom 114 . . . . . . . 8  |-  ( v  e.  B  ->  (
u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B
) ) ) )
1716exlimiv 1534 . . . . . . 7  |-  ( E. v  v  e.  B  ->  ( u  e.  A  ->  u  e.  ran  ( 1st  |`  ( A  X.  B ) ) ) )
1817ssrdv 3031 . . . . . 6  |-  ( E. v  v  e.  B  ->  A  C_  ran  ( 1st  |`  ( A  X.  B
) ) )
19 frn 5169 . . . . . . 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 305 . . . . 5  |-  ( E. v  v  e.  B  ->  ( ran  ( 1st  |`  ( A  X.  B
) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
22 eqss 3040 . . . . 5  |-  ( ran  ( 1st  |`  ( A  X.  B ) )  =  A  <->  ( ran  ( 1st  |`  ( A  X.  B ) )  C_  A  /\  A  C_  ran  ( 1st  |`  ( A  X.  B ) ) ) )
2321, 22sylibr 132 . . . 4  |-  ( E. v  v  e.  B  ->  ran  ( 1st  |`  ( A  X.  B ) )  =  A )
2423, 9jctil 305 . . 3  |-  ( E. v  v  e.  B  ->  ( ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) --> A  /\  ran  ( 1st  |`  ( A  X.  B ) )  =  A ) )
25 dffo2 5237 . . 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 132 . 2  |-  ( E. v  v  e.  B  ->  ( 1st  |`  ( A  X.  B ) ) : ( A  X.  B ) -onto-> A )
272, 26sylbir 133 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 102    = wceq 1289   E.wex 1426    e. wcel 1438    C_ wss 2999   <.cop 3449    X. cxp 4436   ran crn 4439    |` cres 4440    Fn wfn 5010   -->wf 5011   -onto->wfo 5013   ` cfv 5015   1stc1st 5909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-1st 5911
This theorem is referenced by:  1stconst  5986
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