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Theorem 1stconst 5943
Description: The mapping of a restriction of the  1st function to a constant function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
1stconst  |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -1-1-onto-> A )

Proof of Theorem 1stconst
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snmg 3541 . . 3  |-  ( B  e.  V  ->  E. x  x  e.  { B } )
2 fo1stresm 5889 . . 3  |-  ( E. x  x  e.  { B }  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -onto-> A )
31, 2syl 14 . 2  |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -onto-> A )
4 moeq 2781 . . . . . 6  |-  E* x  x  =  <. y ,  B >.
54moani 2015 . . . . 5  |-  E* x
( y  e.  A  /\  x  =  <. y ,  B >. )
6 vex 2618 . . . . . . . 8  |-  y  e. 
_V
76brres 4687 . . . . . . 7  |-  ( x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
8 fo1st 5885 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
9 fofn 5198 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
108, 9ax-mp 7 . . . . . . . . . 10  |-  1st  Fn  _V
11 vex 2618 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5308 . . . . . . . . . 10  |-  ( ( 1st  Fn  _V  /\  x  e.  _V )  ->  ( ( 1st `  x
)  =  y  <->  x 1st y ) )
1310, 11, 12mp2an 417 . . . . . . . . 9  |-  ( ( 1st `  x )  =  y  <->  x 1st y )
1413anbi1i 446 . . . . . . . 8  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
15 elxp7 5898 . . . . . . . . . . 11  |-  ( x  e.  ( A  X.  { B } )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  { B } ) ) )
16 eleq1 2147 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  x )  =  y  ->  (
( 1st `  x
)  e.  A  <->  y  e.  A ) )
1716biimpa 290 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( 1st `  x )  e.  A )  -> 
y  e.  A )
1817adantrr 463 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) )  ->  y  e.  A )
1918adantrl 462 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  -> 
y  e.  A )
20 elsni 3449 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  x )  e.  { B }  ->  ( 2nd `  x
)  =  B )
21 eqopi 5899 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  y  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2221an12s 530 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2320, 22sylanr2 397 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  e.  { B }
) )  ->  x  =  <. y ,  B >. )
2423adantrrl 470 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  ->  x  =  <. y ,  B >. )
2519, 24jca 300 . . . . . . . . . . 11  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  -> 
( y  e.  A  /\  x  =  <. y ,  B >. )
)
2615, 25sylan2b 281 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  ->  ( y  e.  A  /\  x  =  <. y ,  B >. ) )
2726adantl 271 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) ) )  ->  (
y  e.  A  /\  x  =  <. y ,  B >. ) )
28 simprr 499 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  =  <. y ,  B >. )
2928fveq2d 5272 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. y ,  B >. ) )
30 simprl 498 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  y  e.  A )
31 simpl 107 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  V )
32 op1stg 5878 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  V )  ->  ( 1st `  <. y ,  B >. )  =  y )
3330, 31, 32syl2anc 403 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st ` 
<. y ,  B >. )  =  y )
3429, 33eqtrd 2117 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  y )
35 snidg 3456 . . . . . . . . . . . . 13  |-  ( B  e.  V  ->  B  e.  { B } )
3635adantr 270 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  { B } )
37 opelxpi 4442 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  { B } )  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3830, 36, 37syl2anc 403 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3928, 38eqeltrd 2161 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  e.  ( A  X.  { B } ) )
4034, 39jca 300 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( ( 1st `  x )  =  y  /\  x  e.  ( A  X.  { B } ) ) )
4127, 40impbida 561 . . . . . . . 8  |-  ( B  e.  V  ->  (
( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( y  e.  A  /\  x  = 
<. y ,  B >. ) ) )
4214, 41syl5bbr 192 . . . . . . 7  |-  ( B  e.  V  ->  (
( x 1st y  /\  x  e.  ( A  X.  { B }
) )  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
437, 42syl5bb 190 . . . . . 6  |-  ( B  e.  V  ->  (
x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
4443mobidv 1981 . . . . 5  |-  ( B  e.  V  ->  ( E* x  x ( 1st  |`  ( A  X.  { B } ) ) y  <->  E* x ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
455, 44mpbiri 166 . . . 4  |-  ( B  e.  V  ->  E* x  x ( 1st  |`  ( A  X.  { B }
) ) y )
4645alrimiv 1799 . . 3  |-  ( B  e.  V  ->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
47 funcnv2 5039 . . 3  |-  ( Fun  `' ( 1st  |`  ( A  X.  { B }
) )  <->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
4846, 47sylibr 132 . 2  |-  ( B  e.  V  ->  Fun  `' ( 1st  |`  ( A  X.  { B }
) ) )
49 dff1o3 5222 . 2  |-  ( ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B }
)
-1-1-onto-> A 
<->  ( ( 1st  |`  ( A  X.  { B }
) ) : ( A  X.  { B } ) -onto-> A  /\  Fun  `' ( 1st  |`  ( A  X.  { B }
) ) ) )
503, 48, 49sylanbrc 408 1  |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1285    = wceq 1287   E.wex 1424    e. wcel 1436   E*wmo 1946   _Vcvv 2615   {csn 3431   <.cop 3434   class class class wbr 3820    X. cxp 4409   `'ccnv 4410    |` cres 4413   Fun wfun 4975    Fn wfn 4976   -onto->wfo 4979   -1-1-onto->wf1o 4980   ` cfv 4981   1stc1st 5866   2ndc2nd 5867
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-1st 5868  df-2nd 5869
This theorem is referenced by: (None)
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