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Theorem 2ndconst 6788
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 4061 . . 3  |-  ( A  e.  V  ->  { A }  =/=  (/) )
2 fo2ndres 6724 . . 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 3200 . . . . . 6  |-  E* x  x  =  <. A , 
y >.
54moani 2277 . . . . 5  |-  E* x
( y  e.  B  /\  x  =  <. A ,  y >. )
6 vex 3037 . . . . . . . 8  |-  y  e. 
_V
76brres 5192 . . . . . . 7  |-  ( x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
8 fo2nd 6720 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
9 fofn 5705 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
108, 9ax-mp 5 . . . . . . . . . 10  |-  2nd  Fn  _V
11 vex 3037 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5814 . . . . . . . . . 10  |-  ( ( 2nd  Fn  _V  /\  x  e.  _V )  ->  ( ( 2nd `  x
)  =  y  <->  x 2nd y ) )
1310, 11, 12mp2an 670 . . . . . . . . 9  |-  ( ( 2nd `  x )  =  y  <->  x 2nd y )
1413anbi1i 693 . . . . . . . 8  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
15 elxp7 6732 . . . . . . . . . . 11  |-  ( x  e.  ( { A }  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  { A }  /\  ( 2nd `  x )  e.  B ) ) )
16 eleq1 2454 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  x )  =  y  ->  (
( 2nd `  x
)  e.  B  <->  y  e.  B ) )
1716biimpa 482 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( 2nd `  x )  e.  B )  -> 
y  e.  B )
1817adantrl 713 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) )  ->  y  e.  B )
1918adantrl 713 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  y  e.  B )
20 elsni 3969 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x )  e.  { A }  ->  ( 1st `  x
)  =  A )
21 eqopi 6733 . . . . . . . . . . . . . . . 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 651 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  e.  { A }
) )  ->  x  =  <. A ,  y
>. )
2524adantrrr 722 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  x  =  <. A ,  y
>. )
2619, 25jca 530 . . . . . . . . . . 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 473 . . . . . . . . . 10  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  ->  (
y  e.  B  /\  x  =  <. A , 
y >. ) )
2827adantl 464 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) ) )  -> 
( y  e.  B  /\  x  =  <. A ,  y >. )
)
29 fveq2 5774 . . . . . . . . . . . 12  |-  ( x  =  <. A ,  y
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  y
>. ) )
30 op2ndg 6712 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( 2nd `  <. A ,  y >. )  =  y )
316, 30mpan2 669 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( 2nd `  <. A ,  y
>. )  =  y
)
3229, 31sylan9eqr 2445 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  x  =  <. A , 
y >. )  ->  ( 2nd `  x )  =  y )
3332adantrl 713 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( 2nd `  x )  =  y )
34 simprr 755 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  =  <. A ,  y >.
)
35 snidg 3970 . . . . . . . . . . . . 13  |-  ( A  e.  V  ->  A  e.  { A } )
3635adantr 463 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  A  e.  { A } )
37 simprl 754 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  y  e.  B )
38 opelxpi 4945 . . . . . . . . . . . 12  |-  ( ( A  e.  { A }  /\  y  e.  B
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
3936, 37, 38syl2anc 659 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
4034, 39eqeltrd 2470 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  e.  ( { A }  X.  B ) )
4133, 40jca 530 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( ( 2nd `  x )  =  y  /\  x  e.  ( { A }  X.  B ) ) )
4228, 41impbida 830 . . . . . . . 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 2241 . . . . 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 1727 . . 3  |-  ( A  e.  V  ->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
48 funcnv2 5555 . . 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 5730 . 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 662 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 367   A.wal 1397    = wceq 1399    e. wcel 1826   E*wmo 2219    =/= wne 2577   _Vcvv 3034   (/)c0 3711   {csn 3944   <.cop 3950   class class class wbr 4367    X. cxp 4911   `'ccnv 4912    |` cres 4915   Fun wfun 5490    Fn wfn 5491   -onto->wfo 5494   -1-1-onto->wf1o 5495   ` cfv 5496   1stc1st 6697   2ndc2nd 6698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-1st 6699  df-2nd 6700
This theorem is referenced by:  curry1  6791  xpfi  7706  fsum2dlem  13587  fprod2dlem  13786  gsum2dlem2  17112  gsum2dOLD  17114  ovoliunlem1  21998  gsummpt2d  27925
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