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Theorem 2ndconst 6438
Description: The mapping of a restriction of the  2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
2ndconst  |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )

Proof of Theorem 2ndconst
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snnzg 3923 . . 3  |-  ( A  e.  V  ->  { A }  =/=  (/) )
2 fo2ndres 6373 . . 3  |-  ( { A }  =/=  (/)  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -onto-> B )
31, 2syl 16 . 2  |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -onto-> B )
4 moeq 3112 . . . . . 6  |-  E* x  x  =  <. A , 
y >.
54moani 2335 . . . . 5  |-  E* x
( y  e.  B  /\  x  =  <. A ,  y >. )
6 vex 2961 . . . . . . . 8  |-  y  e. 
_V
76brres 5154 . . . . . . 7  |-  ( x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
8 fo2nd 6369 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
9 fofn 5657 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
108, 9ax-mp 8 . . . . . . . . . 10  |-  2nd  Fn  _V
11 vex 2961 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5769 . . . . . . . . . 10  |-  ( ( 2nd  Fn  _V  /\  x  e.  _V )  ->  ( ( 2nd `  x
)  =  y  <->  x 2nd y ) )
1310, 11, 12mp2an 655 . . . . . . . . 9  |-  ( ( 2nd `  x )  =  y  <->  x 2nd y )
1413anbi1i 678 . . . . . . . 8  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
15 elxp7 6381 . . . . . . . . . . 11  |-  ( x  e.  ( { A }  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  { A }  /\  ( 2nd `  x )  e.  B ) ) )
16 eleq1 2498 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  x )  =  y  ->  (
( 2nd `  x
)  e.  B  <->  y  e.  B ) )
1716biimpa 472 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( 2nd `  x )  e.  B )  -> 
y  e.  B )
1817adantrl 698 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) )  ->  y  e.  B )
1918adantrl 698 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  y  e.  B )
20 elsni 3840 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x )  e.  { A }  ->  ( 1st `  x
)  =  A )
21 eqopi 6385 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  A  /\  ( 2nd `  x )  =  y ) )  ->  x  =  <. A ,  y >. )
2221ancom2s 779 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 2nd `  x
)  =  y  /\  ( 1st `  x )  =  A ) )  ->  x  =  <. A ,  y >. )
2322an12s 778 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  =  A ) )  ->  x  =  <. A ,  y >. )
2420, 23sylanr2 636 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  e.  { A }
) )  ->  x  =  <. A ,  y
>. )
2524adantrrr 707 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  x  =  <. A ,  y
>. )
2619, 25jca 520 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  (
y  e.  B  /\  x  =  <. A , 
y >. ) )
2715, 26sylan2b 463 . . . . . . . . . 10  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  ->  (
y  e.  B  /\  x  =  <. A , 
y >. ) )
2827adantl 454 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) ) )  -> 
( y  e.  B  /\  x  =  <. A ,  y >. )
)
29 fveq2 5730 . . . . . . . . . . . 12  |-  ( x  =  <. A ,  y
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  y
>. ) )
30 op2ndg 6362 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( 2nd `  <. A ,  y >. )  =  y )
316, 30mpan2 654 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( 2nd `  <. A ,  y
>. )  =  y
)
3229, 31sylan9eqr 2492 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  x  =  <. A , 
y >. )  ->  ( 2nd `  x )  =  y )
3332adantrl 698 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( 2nd `  x )  =  y )
34 simprr 735 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  =  <. A ,  y >.
)
35 snidg 3841 . . . . . . . . . . . . 13  |-  ( A  e.  V  ->  A  e.  { A } )
3635adantr 453 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  A  e.  { A } )
37 simprl 734 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  y  e.  B )
38 opelxpi 4912 . . . . . . . . . . . 12  |-  ( ( A  e.  { A }  /\  y  e.  B
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
3936, 37, 38syl2anc 644 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
4034, 39eqeltrd 2512 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  e.  ( { A }  X.  B ) )
4133, 40jca 520 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( ( 2nd `  x )  =  y  /\  x  e.  ( { A }  X.  B ) ) )
4228, 41impbida 807 . . . . . . . 8  |-  ( A  e.  V  ->  (
( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
4314, 42syl5bbr 252 . . . . . . 7  |-  ( A  e.  V  ->  (
( x 2nd y  /\  x  e.  ( { A }  X.  B
) )  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
447, 43syl5bb 250 . . . . . 6  |-  ( A  e.  V  ->  (
x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
4544mobidv 2318 . . . . 5  |-  ( A  e.  V  ->  ( E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y  <->  E* x ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
465, 45mpbiri 226 . . . 4  |-  ( A  e.  V  ->  E* x  x ( 2nd  |`  ( { A }  X.  B
) ) y )
4746alrimiv 1642 . . 3  |-  ( A  e.  V  ->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
48 funcnv2 5512 . . 3  |-  ( Fun  `' ( 2nd  |`  ( { A }  X.  B
) )  <->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
4947, 48sylibr 205 . 2  |-  ( A  e.  V  ->  Fun  `' ( 2nd  |`  ( { A }  X.  B
) ) )
50 dff1o3 5682 . 2  |-  ( ( 2nd  |`  ( { A }  X.  B
) ) : ( { A }  X.  B ) -1-1-onto-> B  <->  ( ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -onto-> B  /\  Fun  `' ( 2nd  |`  ( { A }  X.  B
) ) ) )
513, 49, 50sylanbrc 647 1  |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   E*wmo 2284    =/= wne 2601   _Vcvv 2958   (/)c0 3630   {csn 3816   <.cop 3819   class class class wbr 4214    X. cxp 4878   `'ccnv 4879    |` cres 4882   Fun wfun 5450    Fn wfn 5451   -onto->wfo 5454   -1-1-onto->wf1o 5455   ` cfv 5456   1stc1st 6349   2ndc2nd 6350
This theorem is referenced by:  curry1  6440  xpfi  7380  fsum2dlem  12556  gsum2d  15548  ovoliunlem1  19400  fprod2dlem  25306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352
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