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Theorem 2ndconst 5874
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 snmg 3516 . . 3  |-  ( A  e.  V  ->  E. x  x  e.  { A } )
2 fo2ndresm 5820 . . 3  |-  ( E. x  x  e.  { A }  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -onto-> B )
31, 2syl 14 . 2  |-  ( A  e.  V  ->  ( 2nd  |`  ( { A }  X.  B ) ) : ( { A }  X.  B ) -onto-> B )
4 moeq 2768 . . . . . 6  |-  E* x  x  =  <. A , 
y >.
54moani 2012 . . . . 5  |-  E* x
( y  e.  B  /\  x  =  <. A ,  y >. )
6 vex 2605 . . . . . . . 8  |-  y  e. 
_V
76brres 4646 . . . . . . 7  |-  ( x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
8 fo2nd 5816 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
9 fofn 5139 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
108, 9ax-mp 7 . . . . . . . . . 10  |-  2nd  Fn  _V
11 vex 2605 . . . . . . . . . 10  |-  x  e. 
_V
12 fnbrfvb 5246 . . . . . . . . . 10  |-  ( ( 2nd  Fn  _V  /\  x  e.  _V )  ->  ( ( 2nd `  x
)  =  y  <->  x 2nd y ) )
1310, 11, 12mp2an 417 . . . . . . . . 9  |-  ( ( 2nd `  x )  =  y  <->  x 2nd y )
1413anbi1i 446 . . . . . . . 8  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( x 2nd y  /\  x  e.  ( { A }  X.  B ) ) )
15 elxp7 5828 . . . . . . . . . . 11  |-  ( x  e.  ( { A }  X.  B )  <->  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  { A }  /\  ( 2nd `  x )  e.  B ) ) )
16 eleq1 2142 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  x )  =  y  ->  (
( 2nd `  x
)  e.  B  <->  y  e.  B ) )
1716biimpa 290 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( 2nd `  x )  e.  B )  -> 
y  e.  B )
1817adantrl 462 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) )  ->  y  e.  B )
1918adantrl 462 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  y  e.  B )
20 elsni 3424 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x )  e.  { A }  ->  ( 1st `  x
)  =  A )
21 eqopi 5829 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  =  A  /\  ( 2nd `  x )  =  y ) )  ->  x  =  <. A ,  y >. )
2221ancom2s 531 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( _V 
X.  _V )  /\  (
( 2nd `  x
)  =  y  /\  ( 1st `  x )  =  A ) )  ->  x  =  <. A ,  y >. )
2322an12s 530 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  =  A ) )  ->  x  =  <. A ,  y >. )
2420, 23sylanr2 397 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( 1st `  x )  e.  { A }
) )  ->  x  =  <. A ,  y
>. )
2524adantrrr 471 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  x
)  =  y  /\  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  { A }  /\  ( 2nd `  x
)  e.  B ) ) )  ->  x  =  <. A ,  y
>. )
2619, 25jca 300 . . . . . . . . . . 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 281 . . . . . . . . . 10  |-  ( ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  ->  (
y  e.  B  /\  x  =  <. A , 
y >. ) )
2827adantl 271 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) ) )  -> 
( y  e.  B  /\  x  =  <. A ,  y >. )
)
29 fveq2 5209 . . . . . . . . . . . 12  |-  ( x  =  <. A ,  y
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  y
>. ) )
30 op2ndg 5809 . . . . . . . . . . . . 13  |-  ( ( A  e.  V  /\  y  e.  _V )  ->  ( 2nd `  <. A ,  y >. )  =  y )
316, 30mpan2 416 . . . . . . . . . . . 12  |-  ( A  e.  V  ->  ( 2nd `  <. A ,  y
>. )  =  y
)
3229, 31sylan9eqr 2136 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  x  =  <. A , 
y >. )  ->  ( 2nd `  x )  =  y )
3332adantrl 462 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( 2nd `  x )  =  y )
34 simprr 499 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  =  <. A ,  y >.
)
35 snidg 3431 . . . . . . . . . . . . 13  |-  ( A  e.  V  ->  A  e.  { A } )
3635adantr 270 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  A  e.  { A } )
37 simprl 498 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  y  e.  B )
38 opelxpi 4402 . . . . . . . . . . . 12  |-  ( ( A  e.  { A }  /\  y  e.  B
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
3936, 37, 38syl2anc 403 . . . . . . . . . . 11  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  <. A , 
y >.  e.  ( { A }  X.  B
) )
4034, 39eqeltrd 2156 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  x  e.  ( { A }  X.  B ) )
4133, 40jca 300 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ( y  e.  B  /\  x  =  <. A ,  y >. )
)  ->  ( ( 2nd `  x )  =  y  /\  x  e.  ( { A }  X.  B ) ) )
4228, 41impbida 561 . . . . . . . 8  |-  ( A  e.  V  ->  (
( ( 2nd `  x
)  =  y  /\  x  e.  ( { A }  X.  B
) )  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
4314, 42syl5bbr 192 . . . . . . 7  |-  ( A  e.  V  ->  (
( x 2nd y  /\  x  e.  ( { A }  X.  B
) )  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
447, 43syl5bb 190 . . . . . 6  |-  ( A  e.  V  ->  (
x ( 2nd  |`  ( { A }  X.  B
) ) y  <->  ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
4544mobidv 1978 . . . . 5  |-  ( A  e.  V  ->  ( E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y  <->  E* x ( y  e.  B  /\  x  =  <. A ,  y
>. ) ) )
465, 45mpbiri 166 . . . 4  |-  ( A  e.  V  ->  E* x  x ( 2nd  |`  ( { A }  X.  B
) ) y )
4746alrimiv 1796 . . 3  |-  ( A  e.  V  ->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
48 funcnv2 4990 . . 3  |-  ( Fun  `' ( 2nd  |`  ( { A }  X.  B
) )  <->  A. y E* x  x ( 2nd  |`  ( { A }  X.  B ) ) y )
4947, 48sylibr 132 . 2  |-  ( A  e.  V  ->  Fun  `' ( 2nd  |`  ( { A }  X.  B
) ) )
50 dff1o3 5163 . 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 408 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    /\ wa 102    <-> wb 103   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   E*wmo 1943   _Vcvv 2602   {csn 3406   <.cop 3409   class class class wbr 3793    X. cxp 4369   `'ccnv 4370    |` cres 4373   Fun wfun 4926    Fn wfn 4927   -onto->wfo 4930   -1-1-onto->wf1o 4931   ` cfv 4932   1stc1st 5796   2ndc2nd 5797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-1st 5798  df-2nd 5799
This theorem is referenced by: (None)
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