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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 4144 . . 3  |-  ( B  e.  V  ->  { B }  =/=  (/) )
2 fo1stres 6808 . . 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 3279 . . . . . 6  |-  E* x  x  =  <. y ,  B >.
54moani 2348 . . . . 5  |-  E* x
( y  e.  A  /\  x  =  <. y ,  B >. )
6 vex 3116 . . . . . . . 8  |-  y  e. 
_V
76brres 5280 . . . . . . 7  |-  ( x ( 1st  |`  ( A  X.  { B }
) ) y  <->  ( x 1st y  /\  x  e.  ( A  X.  { B } ) ) )
8 fo1st 6804 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
9 fofn 5797 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
108, 9ax-mp 5 . . . . . . . . . 10  |-  1st  Fn  _V
11 vex 3116 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5908 . . . . . . . . . 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 6817 . . . . . . . . . . 11  |-  ( x  e.  ( A  X.  { B } )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x
)  e.  { B } ) ) )
16 eleq1 2539 . . . . . . . . . . . . . . 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 4052 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  x )  e.  { B }  ->  ( 2nd `  x
)  =  B )
21 eqopi 6818 . . . . . . . . . . . . . . 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 5870 . . . . . . . . . . 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 6796 . . . . . . . . . . . 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 2508 . . . . . . . . . 10  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  ( 1st `  x )  =  y )
35 snidg 4053 . . . . . . . . . . . . 13  |-  ( B  e.  V  ->  B  e.  { B } )
3635adantr 465 . . . . . . . . . . . 12  |-  ( ( B  e.  V  /\  ( y  e.  A  /\  x  =  <. y ,  B >. )
)  ->  B  e.  { B } )
37 opelxpi 5031 . . . . . . . . . . . 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 2555 . . . . . . . . . 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 830 . . . . . . . 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 2299 . . . . 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 1695 . . 3  |-  ( B  e.  V  ->  A. y E* x  x ( 1st  |`  ( A  X.  { B } ) ) y )
47 funcnv2 5647 . . 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 5822 . 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 1377    = wceq 1379    e. wcel 1767   E*wmo 2276    =/= wne 2662   _Vcvv 3113   (/)c0 3785   {csn 4027   <.cop 4033   class class class wbr 4447    X. cxp 4997   `'ccnv 4998    |` cres 5001   Fun wfun 5582    Fn wfn 5583   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588   1stc1st 6782   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1st 6784  df-2nd 6785
This theorem is referenced by:  curry2  6878  domss2  7676
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