MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsc1 Structured version   Unicode version

Theorem xpsc1 15466
Description: The pair function maps  1 to  B. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
xpsc1  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )

Proof of Theorem xpsc1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xpsc 15462 . . . 4  |-  `' ( { A }  +c  { B } )  =  ( ( { (/) }  X.  { A }
)  u.  ( { 1o }  X.  { B } ) )
21fveq1i 5882 . . 3  |-  ( `' ( { A }  +c  { B } ) `
 1o )  =  ( ( ( {
(/) }  X.  { A } )  u.  ( { 1o }  X.  { B } ) ) `  1o )
3 vex 3083 . . . . . . . . . . . . 13  |-  x  e. 
_V
4 fvi 5938 . . . . . . . . . . . . 13  |-  ( x  e.  _V  ->  (  _I  `  x )  =  x )
53, 4ax-mp 5 . . . . . . . . . . . 12  |-  (  _I 
`  x )  =  x
6 elsni 4023 . . . . . . . . . . . . 13  |-  ( x  e.  { A }  ->  x  =  A )
76fveq2d 5885 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  (  _I  `  x
)  =  (  _I 
`  A ) )
85, 7syl5eqr 2477 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  x  =  (  _I 
`  A ) )
9 elsn 4012 . . . . . . . . . . 11  |-  ( x  e.  { (  _I 
`  A ) }  <-> 
x  =  (  _I 
`  A ) )
108, 9sylibr 215 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  x  e.  { (  _I  `  A ) } )
1110ssriv 3468 . . . . . . . . 9  |-  { A }  C_  { (  _I 
`  A ) }
12 xpss2 4963 . . . . . . . . 9  |-  ( { A }  C_  { (  _I  `  A ) }  ->  ( { (/)
}  X.  { A } )  C_  ( { (/) }  X.  {
(  _I  `  A
) } ) )
1311, 12ax-mp 5 . . . . . . . 8  |-  ( {
(/) }  X.  { A } )  C_  ( { (/) }  X.  {
(  _I  `  A
) } )
14 0ex 4556 . . . . . . . . 9  |-  (/)  e.  _V
15 fvex 5891 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
1614, 15xpsn 6081 . . . . . . . 8  |-  ( {
(/) }  X.  { (  _I  `  A ) } )  =  { <.
(/) ,  (  _I  `  A ) >. }
1713, 16sseqtri 3496 . . . . . . 7  |-  ( {
(/) }  X.  { A } )  C_  { <. (/)
,  (  _I  `  A ) >. }
1814, 15funsn 5649 . . . . . . 7  |-  Fun  { <.
(/) ,  (  _I  `  A ) >. }
19 funss 5619 . . . . . . 7  |-  ( ( { (/) }  X.  { A } )  C_  { <. (/)
,  (  _I  `  A ) >. }  ->  ( Fun  { <. (/) ,  (  _I  `  A )
>. }  ->  Fun  ( {
(/) }  X.  { A } ) ) )
2017, 18, 19mp2 9 . . . . . 6  |-  Fun  ( { (/) }  X.  { A } )
21 funfn 5630 . . . . . 6  |-  ( Fun  ( { (/) }  X.  { A } )  <->  ( { (/)
}  X.  { A } )  Fn  dom  ( { (/) }  X.  { A } ) )
2220, 21mpbi 211 . . . . 5  |-  ( {
(/) }  X.  { A } )  Fn  dom  ( { (/) }  X.  { A } )
2322a1i 11 . . . 4  |-  ( B  e.  V  ->  ( { (/) }  X.  { A } )  Fn  dom  ( { (/) }  X.  { A } ) )
24 fnconstg 5788 . . . 4  |-  ( B  e.  V  ->  ( { 1o }  X.  { B } )  Fn  { 1o } )
25 dmxpss 5287 . . . . . . 7  |-  dom  ( { (/) }  X.  { A } )  C_  { (/) }
26 ssrin 3687 . . . . . . 7  |-  ( dom  ( { (/) }  X.  { A } )  C_  {
(/) }  ->  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } ) 
C_  ( { (/) }  i^i  { 1o }
) )
2725, 26ax-mp 5 . . . . . 6  |-  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } ) 
C_  ( { (/) }  i^i  { 1o }
)
28 1n0 7208 . . . . . . . 8  |-  1o  =/=  (/)
2928necomi 2690 . . . . . . 7  |-  (/)  =/=  1o
30 disjsn2 4061 . . . . . . 7  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
3129, 30ax-mp 5 . . . . . 6  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
32 sseq0 3796 . . . . . 6  |-  ( ( ( dom  ( {
(/) }  X.  { A } )  i^i  { 1o } )  C_  ( { (/) }  i^i  { 1o } )  /\  ( { (/) }  i^i  { 1o } )  =  (/) )  ->  ( dom  ( { (/) }  X.  { A } )  i^i  { 1o } )  =  (/) )
3327, 31, 32mp2an 676 . . . . 5  |-  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } )  =  (/)
3433a1i 11 . . . 4  |-  ( B  e.  V  ->  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } )  =  (/) )
35 1on 7200 . . . . . . 7  |-  1o  e.  On
3635elexi 3090 . . . . . 6  |-  1o  e.  _V
3736snid 4026 . . . . 5  |-  1o  e.  { 1o }
3837a1i 11 . . . 4  |-  ( B  e.  V  ->  1o  e.  { 1o } )
39 fvun2 5953 . . . 4  |-  ( ( ( { (/) }  X.  { A } )  Fn 
dom  ( { (/) }  X.  { A }
)  /\  ( { 1o }  X.  { B } )  Fn  { 1o }  /\  ( ( dom  ( { (/) }  X.  { A }
)  i^i  { 1o } )  =  (/)  /\  1o  e.  { 1o } ) )  -> 
( ( ( {
(/) }  X.  { A } )  u.  ( { 1o }  X.  { B } ) ) `  1o )  =  (
( { 1o }  X.  { B } ) `
 1o ) )
4023, 24, 34, 38, 39syl112anc 1268 . . 3  |-  ( B  e.  V  ->  (
( ( { (/) }  X.  { A }
)  u.  ( { 1o }  X.  { B } ) ) `  1o )  =  (
( { 1o }  X.  { B } ) `
 1o ) )
412, 40syl5eq 2475 . 2  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  ( ( { 1o }  X.  { B } ) `  1o ) )
42 xpsng 6080 . . . . 5  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( { 1o }  X.  { B } )  =  { <. 1o ,  B >. } )
4342fveq1d 5883 . . . 4  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( ( { 1o }  X.  { B }
) `  1o )  =  ( { <. 1o ,  B >. } `  1o ) )
44 fvsng 6113 . . . 4  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( { <. 1o ,  B >. } `  1o )  =  B )
4543, 44eqtrd 2463 . . 3  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( ( { 1o }  X.  { B }
) `  1o )  =  B )
4635, 45mpan 674 . 2  |-  ( B  e.  V  ->  (
( { 1o }  X.  { B } ) `
 1o )  =  B )
4741, 46eqtrd 2463 1  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   _Vcvv 3080    u. cun 3434    i^i cin 3435    C_ wss 3436   (/)c0 3761   {csn 3998   <.cop 4004    _I cid 4763    X. cxp 4851   `'ccnv 4852   dom cdm 4853   Oncon0 5442   Fun wfun 5595    Fn wfn 5596   ` cfv 5601  (class class class)co 6305   1oc1o 7186    +c ccda 8604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1o 7193  df-cda 8605
This theorem is referenced by:  xpscfv  15467  xpsfeq  15469  xpsfrnel2  15470  xpsff1o  15473  xpsle  15486  dmdprdpr  17681  dprdpr  17682  xpstopnlem1  20822  xpstopnlem2  20824  xpsxmetlem  21392  xpsdsval  21394  xpsmet  21395
  Copyright terms: Public domain W3C validator