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Theorem 2ndconst 6677
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 4007 . . 3  |-  ( A  e.  V  ->  { A }  =/=  (/) )
2 fo2ndres 6616 . . 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 3150 . . . . . 6  |-  E* x  x  =  <. A , 
y >.
54moani 2326 . . . . 5  |-  E* x
( y  e.  B  /\  x  =  <. A ,  y >. )
6 vex 2990 . . . . . . . 8  |-  y  e. 
_V
76brres 5132 . . . . . . 7  |-  ( x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
8 fo2nd 6612 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
9 fofn 5637 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
108, 9ax-mp 5 . . . . . . . . . 10  |-  2nd  Fn  _V
11 vex 2990 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5747 . . . . . . . . . 10  |-  ( ( 2nd  Fn  _V  /\  x  e.  _V )  ->  ( ( 2nd `  x
)  =  y  <->  x 2nd y ) )
1310, 11, 12mp2an 672 . . . . . . . . 9  |-  ( ( 2nd `  x )  =  y  <->  x 2nd y )
1413anbi1i 695 . . . . . . . 8  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
15 elxp7 6624 . . . . . . . . . . 11  |-  ( x  e.  ( { A }  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  { A }  /\  ( 2nd `  x )  e.  B ) ) )
16 eleq1 2503 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  x )  =  y  ->  (
( 2nd `  x
)  e.  B  <->  y  e.  B ) )
1716biimpa 484 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( 2nd `  x )  e.  B )  -> 
y  e.  B )
1817adantrl 715 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) )  ->  y  e.  B )
1918adantrl 715 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  y  e.  B )
20 elsni 3917 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x )  e.  { A }  ->  ( 1st `  x
)  =  A )
21 eqopi 6625 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  A  /\  ( 2nd `  x )  =  y ) )  ->  x  =  <. A ,  y >. )
2221ancom2s 800 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 2nd `  x
)  =  y  /\  ( 1st `  x )  =  A ) )  ->  x  =  <. A ,  y >. )
2322an12s 799 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  =  A ) )  ->  x  =  <. A ,  y >. )
2420, 23sylanr2 653 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  e.  { A }
) )  ->  x  =  <. A ,  y
>. )
2524adantrrr 724 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  x  =  <. A ,  y
>. )
2619, 25jca 532 . . . . . . . . . . 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 475 . . . . . . . . . 10  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  ->  (
y  e.  B  /\  x  =  <. A , 
y >. ) )
2827adantl 466 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) ) )  -> 
( y  e.  B  /\  x  =  <. A ,  y >. )
)
29 fveq2 5706 . . . . . . . . . . . 12  |-  ( x  =  <. A ,  y
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  y
>. ) )
30 op2ndg 6605 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( 2nd `  <. A ,  y >. )  =  y )
316, 30mpan2 671 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( 2nd `  <. A ,  y
>. )  =  y
)
3229, 31sylan9eqr 2497 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  x  =  <. A , 
y >. )  ->  ( 2nd `  x )  =  y )
3332adantrl 715 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( 2nd `  x )  =  y )
34 simprr 756 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  =  <. A ,  y >.
)
35 snidg 3918 . . . . . . . . . . . . 13  |-  ( A  e.  V  ->  A  e.  { A } )
3635adantr 465 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  A  e.  { A } )
37 simprl 755 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  y  e.  B )
38 opelxpi 4886 . . . . . . . . . . . 12  |-  ( ( A  e.  { A }  /\  y  e.  B
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
3936, 37, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
4034, 39eqeltrd 2517 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  e.  ( { A }  X.  B ) )
4133, 40jca 532 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( ( 2nd `  x )  =  y  /\  x  e.  ( { A }  X.  B ) ) )
4228, 41impbida 828 . . . . . . . 8  |-  ( A  e.  V  ->  (
( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
4314, 42syl5bbr 259 . . . . . . 7  |-  ( A  e.  V  ->  (
( x 2nd y  /\  x  e.  ( { A }  X.  B
) )  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
447, 43syl5bb 257 . . . . . 6  |-  ( A  e.  V  ->  (
x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
4544mobidv 2277 . . . . 5  |-  ( A  e.  V  ->  ( E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y  <->  E* x ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
465, 45mpbiri 233 . . . 4  |-  ( A  e.  V  ->  E* x  x ( 2nd  |`  ( { A }  X.  B
) ) y )
4746alrimiv 1685 . . 3  |-  ( A  e.  V  ->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
48 funcnv2 5492 . . 3  |-  ( Fun  `' ( 2nd  |`  ( { A }  X.  B
) )  <->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
4947, 48sylibr 212 . 2  |-  ( A  e.  V  ->  Fun  `' ( 2nd  |`  ( { A }  X.  B
) ) )
50 dff1o3 5662 . 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 664 1  |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -1-1-onto-> B )
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 2620   _Vcvv 2987   (/)c0 3652   {csn 3892   <.cop 3898   class class class wbr 4307    X. cxp 4853   `'ccnv 4854    |` cres 4857   Fun wfun 5427    Fn wfn 5428   -onto->wfo 5431   -1-1-onto->wf1o 5432   ` cfv 5433   1stc1st 6590   2ndc2nd 6591
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 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
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 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-1st 6592  df-2nd 6593
This theorem is referenced by:  curry1  6679  xpfi  7598  fsum2dlem  13252  gsum2dlem2  16477  gsum2dOLD  16479  ovoliunlem1  21000  fprod2dlem  27506
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