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Theorem xpsc1 14495
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 14491 . . . 4  |-  `' ( { A }  +c  { B } )  =  ( ( { (/) }  X.  { A }
)  u.  ( { 1o }  X.  { B } ) )
21fveq1i 5689 . . 3  |-  ( `' ( { A }  +c  { B } ) `
 1o )  =  ( ( ( {
(/) }  X.  { A } )  u.  ( { 1o }  X.  { B } ) ) `  1o )
3 vex 2973 . . . . . . . . . . . . 13  |-  x  e. 
_V
4 fvi 5745 . . . . . . . . . . . . 13  |-  ( x  e.  _V  ->  (  _I  `  x )  =  x )
53, 4ax-mp 5 . . . . . . . . . . . 12  |-  (  _I 
`  x )  =  x
6 elsni 3899 . . . . . . . . . . . . 13  |-  ( x  e.  { A }  ->  x  =  A )
76fveq2d 5692 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  (  _I  `  x
)  =  (  _I 
`  A ) )
85, 7syl5eqr 2487 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  x  =  (  _I 
`  A ) )
9 elsn 3888 . . . . . . . . . . 11  |-  ( x  e.  { (  _I 
`  A ) }  <-> 
x  =  (  _I 
`  A ) )
108, 9sylibr 212 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  x  e.  { (  _I  `  A ) } )
1110ssriv 3357 . . . . . . . . 9  |-  { A }  C_  { (  _I 
`  A ) }
12 xpss2 4945 . . . . . . . . 9  |-  ( { A }  C_  { (  _I  `  A ) }  ->  ( { (/)
}  X.  { A } )  C_  ( { (/) }  X.  {
(  _I  `  A
) } ) )
1311, 12ax-mp 5 . . . . . . . 8  |-  ( {
(/) }  X.  { A } )  C_  ( { (/) }  X.  {
(  _I  `  A
) } )
14 0ex 4419 . . . . . . . . 9  |-  (/)  e.  _V
15 fvex 5698 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
1614, 15xpsn 5882 . . . . . . . 8  |-  ( {
(/) }  X.  { (  _I  `  A ) } )  =  { <.
(/) ,  (  _I  `  A ) >. }
1713, 16sseqtri 3385 . . . . . . 7  |-  ( {
(/) }  X.  { A } )  C_  { <. (/)
,  (  _I  `  A ) >. }
1814, 15funsn 5463 . . . . . . 7  |-  Fun  { <.
(/) ,  (  _I  `  A ) >. }
19 funss 5433 . . . . . . 7  |-  ( ( { (/) }  X.  { A } )  C_  { <. (/)
,  (  _I  `  A ) >. }  ->  ( Fun  { <. (/) ,  (  _I  `  A )
>. }  ->  Fun  ( {
(/) }  X.  { A } ) ) )
2017, 18, 19mp2 9 . . . . . 6  |-  Fun  ( { (/) }  X.  { A } )
21 funfn 5444 . . . . . 6  |-  ( Fun  ( { (/) }  X.  { A } )  <->  ( { (/)
}  X.  { A } )  Fn  dom  ( { (/) }  X.  { A } ) )
2220, 21mpbi 208 . . . . 5  |-  ( {
(/) }  X.  { A } )  Fn  dom  ( { (/) }  X.  { A } )
2322a1i 11 . . . 4  |-  ( B  e.  V  ->  ( { (/) }  X.  { A } )  Fn  dom  ( { (/) }  X.  { A } ) )
24 fnconstg 5595 . . . 4  |-  ( B  e.  V  ->  ( { 1o }  X.  { B } )  Fn  { 1o } )
25 dmxpss 5266 . . . . . . 7  |-  dom  ( { (/) }  X.  { A } )  C_  { (/) }
26 ssrin 3572 . . . . . . 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 6931 . . . . . . . 8  |-  1o  =/=  (/)
2928necomi 2692 . . . . . . 7  |-  (/)  =/=  1o
30 disjsn2 3934 . . . . . . 7  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
3129, 30ax-mp 5 . . . . . 6  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
32 sseq0 3666 . . . . . 6  |-  ( ( ( dom  ( {
(/) }  X.  { A } )  i^i  { 1o } )  C_  ( { (/) }  i^i  { 1o } )  /\  ( { (/) }  i^i  { 1o } )  =  (/) )  ->  ( dom  ( { (/) }  X.  { A } )  i^i  { 1o } )  =  (/) )
3327, 31, 32mp2an 667 . . . . 5  |-  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } )  =  (/)
3433a1i 11 . . . 4  |-  ( B  e.  V  ->  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } )  =  (/) )
35 1on 6923 . . . . . . 7  |-  1o  e.  On
3635elexi 2980 . . . . . 6  |-  1o  e.  _V
3736snid 3902 . . . . 5  |-  1o  e.  { 1o }
3837a1i 11 . . . 4  |-  ( B  e.  V  ->  1o  e.  { 1o } )
39 fvun2 5760 . . . 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 1217 . . 3  |-  ( B  e.  V  ->  (
( ( { (/) }  X.  { A }
)  u.  ( { 1o }  X.  { B } ) ) `  1o )  =  (
( { 1o }  X.  { B } ) `
 1o ) )
412, 40syl5eq 2485 . 2  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  ( ( { 1o }  X.  { B } ) `  1o ) )
42 xpsng 5881 . . . . 5  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( { 1o }  X.  { B } )  =  { <. 1o ,  B >. } )
4342fveq1d 5690 . . . 4  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( ( { 1o }  X.  { B }
) `  1o )  =  ( { <. 1o ,  B >. } `  1o ) )
44 fvsng 5909 . . . 4  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( { <. 1o ,  B >. } `  1o )  =  B )
4543, 44eqtrd 2473 . . 3  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( ( { 1o }  X.  { B }
) `  1o )  =  B )
4635, 45mpan 665 . 2  |-  ( B  e.  V  ->  (
( { 1o }  X.  { B } ) `
 1o )  =  B )
4741, 46eqtrd 2473 1  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   <.cop 3880    _I cid 4627   Oncon0 4715    X. cxp 4834   `'ccnv 4835   dom cdm 4836   Fun wfun 5409    Fn wfn 5410   ` cfv 5415  (class class class)co 6090   1oc1o 6909    +c ccda 8332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1o 6916  df-cda 8333
This theorem is referenced by:  xpscfv  14496  xpsfeq  14498  xpsfrnel2  14499  xpsff1o  14502  xpsle  14515  dmdprdpr  16538  dprdpr  16539  xpstopnlem1  19341  xpstopnlem2  19343  xpsxmetlem  19913  xpsdsval  19915  xpsmet  19916
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