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Theorem 1stconst 6871
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 4057 . . 3  |-  ( B  e.  V  ->  { B }  =/=  (/) )
2 fo1stres 6804 . . 3  |-  ( { B }  =/=  (/)  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -onto-> A )
31, 2syl 17 . 2  |-  ( B  e.  V  ->  ( 1st  |`  ( A  X.  { B } ) ) : ( A  X.  { B } ) -onto-> A )
4 moeq 3181 . . . . . 6  |-  E* x  x  =  <. y ,  B >.
54moani 2354 . . . . 5  |-  E* x
( y  e.  A  /\  x  =  <. y ,  B >. )
6 vex 3015 . . . . . . . 8  |-  y  e. 
_V
76brres 5088 . . . . . . 7  |-  ( x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
8 fo1st 6800 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
9 fofn 5777 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
108, 9ax-mp 5 . . . . . . . . . 10  |-  1st  Fn  _V
11 vex 3015 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5887 . . . . . . . . . 10  |-  ( ( 1st  Fn  _V  /\  x  e.  _V )  ->  ( ( 1st `  x
)  =  y  <->  x 1st y ) )
1310, 11, 12mp2an 683 . . . . . . . . 9  |-  ( ( 1st `  x )  =  y  <->  x 1st y )
1413anbi1i 706 . . . . . . . 8  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
15 elxp7 6813 . . . . . . . . . . 11  |-  ( x  e.  ( A  X.  { B } )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  { B } ) ) )
16 eleq1 2517 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  x )  =  y  ->  (
( 1st `  x
)  e.  A  <->  y  e.  A ) )
1716biimpa 491 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( 1st `  x )  e.  A )  -> 
y  e.  A )
1817adantrr 728 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) )  ->  y  e.  A )
1918adantrl 727 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  -> 
y  e.  A )
20 elsni 3960 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  x )  e.  { B }  ->  ( 2nd `  x
)  =  B )
21 eqopi 6814 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  y  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2221an12s 815 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  =  B ) )  ->  x  =  <. y ,  B >. )
2320, 22sylanr2 663 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 2nd `  x )  e.  { B }
) )  ->  x  =  <. y ,  B >. )
2423adantrrl 735 . . . . . . . . . . . 12  |-  ( ( ( 1st `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  { B }
) ) )  ->  x  =  <. y ,  B >. )
2519, 24jca 539 . . . . . . . . . . 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 482 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  ->  ( y  e.  A  /\  x  =  <. y ,  B >. ) )
2726adantl 472 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) ) )  ->  (
y  e.  A  /\  x  =  <. y ,  B >. ) )
28 simprr 771 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  =  <. y ,  B >. )
2928fveq2d 5851 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. y ,  B >. ) )
30 simprl 769 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  y  e.  A )
31 simpl 463 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  V )
32 op1stg 6792 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  V )  ->  ( 1st `  <. y ,  B >. )  =  y )
3330, 31, 32syl2anc 671 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st ` 
<. y ,  B >. )  =  y )
3429, 33eqtrd 2485 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  y )
35 snidg 3961 . . . . . . . . . . . . 13  |-  ( B  e.  V  ->  B  e.  { B } )
3635adantr 471 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  { B } )
37 opelxpi 4843 . . . . . . . . . . . 12  |-  ( ( y  e.  A  /\  B  e.  { B } )  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3830, 36, 37syl2anc 671 . . . . . . . . . . 11  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  <. y ,  B >.  e.  ( A  X.  { B }
) )
3928, 38eqeltrd 2529 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  x  e.  ( A  X.  { B } ) )
4034, 39jca 539 . . . . . . . . 9  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( ( 1st `  x )  =  y  /\  x  e.  ( A  X.  { B } ) ) )
4127, 40impbida 847 . . . . . . . 8  |-  ( B  e.  V  ->  (
( ( 1st `  x
)  =  y  /\  x  e.  ( A  X.  { B } ) )  <->  ( y  e.  A  /\  x  = 
<. y ,  B >. ) ) )
4214, 41syl5bbr 267 . . . . . . 7  |-  ( B  e.  V  ->  (
( x 1st y  /\  x  e.  ( A  X.  { B }
) )  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
437, 42syl5bb 265 . . . . . 6  |-  ( B  e.  V  ->  (
x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
4443mobidv 2320 . . . . 5  |-  ( B  e.  V  ->  ( E* x  x ( 1st  |`  ( A  X.  { B } ) ) y  <->  E* x ( y  e.  A  /\  x  =  <. y ,  B >. ) ) )
455, 44mpbiri 241 . . . 4  |-  ( B  e.  V  ->  E* x  x ( 1st  |`  ( A  X.  { B }
) ) y )
4645alrimiv 1776 . . 3  |-  ( B  e.  V  ->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
47 funcnv2 5623 . . 3  |-  ( Fun  `' ( 1st  |`  ( A  X.  { B }
) )  <->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
4846, 47sylibr 217 . 2  |-  ( B  e.  V  ->  Fun  `' ( 1st  |`  ( A  X.  { B }
) ) )
49 dff1o3 5802 . 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 675 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 189    /\ wa 375   A.wal 1445    = wceq 1447    e. wcel 1890   E*wmo 2300    =/= wne 2621   _Vcvv 3012   (/)c0 3698   {csn 3935   <.cop 3941   class class class wbr 4373    X. cxp 4809   `'ccnv 4810    |` cres 4813   Fun wfun 5554    Fn wfn 5555   -onto->wfo 5558   -1-1-onto->wf1o 5559   ` cfv 5560   1stc1st 6778   2ndc2nd 6779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-id 4726  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-1st 6780  df-2nd 6781
This theorem is referenced by:  curry2  6878  domss2  7717  fv1stcnv  30427
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