MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stconst Structured version   Unicode version

Theorem 1stconst 6666
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 snnzg 3997 . . 3  |-  ( B  e.  V  ->  { B }  =/=  (/) )
2 fo1stres 6605 . . 3  |-  ( { B }  =/=  (/)  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -onto-> A )
31, 2syl 16 . 2  |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -onto-> A )
4 moeq 3140 . . . . . 6  |-  E* x  x  =  <. y ,  B >.
54moani 2326 . . . . 5  |-  E* x
( y  e.  A  /\  x  =  <. y ,  B >. )
6 vex 2980 . . . . . . . 8  |-  y  e. 
_V
76brres 5122 . . . . . . 7  |-  ( x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
8 fo1st 6601 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
9 fofn 5627 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
108, 9ax-mp 5 . . . . . . . . . 10  |-  1st  Fn  _V
11 vex 2980 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5737 . . . . . . . . . 10  |-  ( ( 1st  Fn  _V  /\  x  e.  _V )  ->  ( ( 1st `  x
)  =  y  <->  x 1st y ) )
1310, 11, 12mp2an 672 . . . . . . . . 9  |-  ( ( 1st `  x )  =  y  <->  x 1st y )
1413anbi1i 695 . . . . . . . 8  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
15 elxp7 6614 . . . . . . . . . . 11  |-  ( x  e.  ( A  X.  { B } )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  { B } ) ) )
16 eleq1 2503 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  x )  =  y  ->  (
( 1st `  x
)  e.  A  <->  y  e.  A ) )
1716biimpa 484 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( 1st `  x )  e.  A )  -> 
y  e.  A )
1817adantrr 716 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) )  ->  y  e.  A )
1918adantrl 715 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  -> 
y  e.  A )
20 elsni 3907 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  x )  e.  { B }  ->  ( 2nd `  x
)  =  B )
21 eqopi 6615 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  y  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2221an12s 799 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2320, 22sylanr2 653 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  e.  { B }
) )  ->  x  =  <. y ,  B >. )
2423adantrrl 723 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  ->  x  =  <. y ,  B >. )
2519, 24jca 532 . . . . . . . . . . 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 475 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  ->  ( y  e.  A  /\  x  =  <. y ,  B >. ) )
2726adantl 466 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) ) )  ->  (
y  e.  A  /\  x  =  <. y ,  B >. ) )
28 simprr 756 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  =  <. y ,  B >. )
2928fveq2d 5700 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. y ,  B >. ) )
30 simprl 755 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  y  e.  A )
31 simpl 457 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  V )
32 op1stg 6594 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  V )  ->  ( 1st `  <. y ,  B >. )  =  y )
3330, 31, 32syl2anc 661 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st ` 
<. y ,  B >. )  =  y )
3429, 33eqtrd 2475 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  y )
35 snidg 3908 . . . . . . . . . . . . 13  |-  ( B  e.  V  ->  B  e.  { B } )
3635adantr 465 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  { B } )
37 opelxpi 4876 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  { B } )  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3830, 36, 37syl2anc 661 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3928, 38eqeltrd 2517 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  e.  ( A  X.  { B } ) )
4034, 39jca 532 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( ( 1st `  x )  =  y  /\  x  e.  ( A  X.  { B } ) ) )
4127, 40impbida 828 . . . . . . . 8  |-  ( B  e.  V  ->  (
( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( y  e.  A  /\  x  = 
<. y ,  B >. ) ) )
4214, 41syl5bbr 259 . . . . . . 7  |-  ( B  e.  V  ->  (
( x 1st y  /\  x  e.  ( A  X.  { B }
) )  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
437, 42syl5bb 257 . . . . . 6  |-  ( B  e.  V  ->  (
x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
4443mobidv 2277 . . . . 5  |-  ( B  e.  V  ->  ( E* x  x ( 1st  |`  ( A  X.  { B } ) ) y  <->  E* x ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
455, 44mpbiri 233 . . . 4  |-  ( B  e.  V  ->  E* x  x ( 1st  |`  ( A  X.  { B }
) ) y )
4645alrimiv 1685 . . 3  |-  ( B  e.  V  ->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
47 funcnv2 5482 . . 3  |-  ( Fun  `' ( 1st  |`  ( A  X.  { B }
) )  <->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
4846, 47sylibr 212 . 2  |-  ( B  e.  V  ->  Fun  `' ( 1st  |`  ( A  X.  { B }
) ) )
49 dff1o3 5652 . 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 664 1  |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -1-1-onto-> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   E*wmo 2254    =/= wne 2611   _Vcvv 2977   (/)c0 3642   {csn 3882   <.cop 3888   class class class wbr 4297    X. cxp 4843   `'ccnv 4844    |` cres 4847   Fun wfun 5417    Fn wfn 5418   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423   1stc1st 6580   2ndc2nd 6581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-1st 6582  df-2nd 6583
This theorem is referenced by:  curry2  6672  domss2  7475
  Copyright terms: Public domain W3C validator