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Theorem 1stconst 6212
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 3707 . . 3  |-  ( B  e.  V  ->  E. x  x  e.  { B } )
2 fo1stresm 6152 . . 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 2910 . . . . . 6  |-  E* x  x  =  <. y ,  B >.
54moani 2094 . . . . 5  |-  E* x
( y  e.  A  /\  x  =  <. y ,  B >. )
6 vex 2738 . . . . . . . 8  |-  y  e. 
_V
76brres 4906 . . . . . . 7  |-  ( x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
8 fo1st 6148 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
9 fofn 5432 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
108, 9ax-mp 5 . . . . . . . . . 10  |-  1st  Fn  _V
11 vex 2738 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5548 . . . . . . . . . 10  |-  ( ( 1st  Fn  _V  /\  x  e.  _V )  ->  ( ( 1st `  x
)  =  y  <->  x 1st y ) )
1310, 11, 12mp2an 426 . . . . . . . . 9  |-  ( ( 1st `  x )  =  y  <->  x 1st y )
1413anbi1i 458 . . . . . . . 8  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
15 elxp7 6161 . . . . . . . . . . 11  |-  ( x  e.  ( A  X.  { B } )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  { B } ) ) )
16 eleq1 2238 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  x )  =  y  ->  (
( 1st `  x
)  e.  A  <->  y  e.  A ) )
1716biimpa 296 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( 1st `  x )  e.  A )  -> 
y  e.  A )
1817adantrr 479 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) )  ->  y  e.  A )
1918adantrl 478 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  -> 
y  e.  A )
20 elsni 3607 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  x )  e.  { B }  ->  ( 2nd `  x
)  =  B )
21 eqopi 6163 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  y  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2221an12s 565 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2320, 22sylanr2 405 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  e.  { B }
) )  ->  x  =  <. y ,  B >. )
2423adantrrl 486 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  ->  x  =  <. y ,  B >. )
2519, 24jca 306 . . . . . . . . . . 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 287 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  ->  ( y  e.  A  /\  x  =  <. y ,  B >. ) )
2726adantl 277 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) ) )  ->  (
y  e.  A  /\  x  =  <. y ,  B >. ) )
28 simprr 531 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  =  <. y ,  B >. )
2928fveq2d 5511 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. y ,  B >. ) )
30 simprl 529 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  y  e.  A )
31 simpl 109 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  V )
32 op1stg 6141 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  V )  ->  ( 1st `  <. y ,  B >. )  =  y )
3330, 31, 32syl2anc 411 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st ` 
<. y ,  B >. )  =  y )
3429, 33eqtrd 2208 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  y )
35 snidg 3618 . . . . . . . . . . . . 13  |-  ( B  e.  V  ->  B  e.  { B } )
3635adantr 276 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  { B } )
37 opelxpi 4652 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  { B } )  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3830, 36, 37syl2anc 411 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3928, 38eqeltrd 2252 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  e.  ( A  X.  { B } ) )
4034, 39jca 306 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( ( 1st `  x )  =  y  /\  x  e.  ( A  X.  { B } ) ) )
4127, 40impbida 596 . . . . . . . 8  |-  ( B  e.  V  ->  (
( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( y  e.  A  /\  x  = 
<. y ,  B >. ) ) )
4214, 41bitr3id 194 . . . . . . 7  |-  ( B  e.  V  ->  (
( x 1st y  /\  x  e.  ( A  X.  { B }
) )  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
437, 42bitrid 192 . . . . . 6  |-  ( B  e.  V  ->  (
x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
4443mobidv 2060 . . . . 5  |-  ( B  e.  V  ->  ( E* x  x ( 1st  |`  ( A  X.  { B } ) ) y  <->  E* x ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
455, 44mpbiri 168 . . . 4  |-  ( B  e.  V  ->  E* x  x ( 1st  |`  ( A  X.  { B }
) ) y )
4645alrimiv 1872 . . 3  |-  ( B  e.  V  ->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
47 funcnv2 5268 . . 3  |-  ( Fun  `' ( 1st  |`  ( A  X.  { B }
) )  <->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
4846, 47sylibr 134 . 2  |-  ( B  e.  V  ->  Fun  `' ( 1st  |`  ( A  X.  { B }
) ) )
49 dff1o3 5459 . 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 417 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 104    <-> wb 105   A.wal 1351    = wceq 1353   E.wex 1490   E*wmo 2025    e. wcel 2146   _Vcvv 2735   {csn 3589   <.cop 3592   class class class wbr 3998    X. cxp 4618   `'ccnv 4619    |` cres 4622   Fun wfun 5202    Fn wfn 5203   -onto->wfo 5206   -1-1-onto->wf1o 5207   ` cfv 5208   1stc1st 6129   2ndc2nd 6130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132
This theorem is referenced by: (None)
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