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Theorem xpsc1 14940
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 14936 . . . 4  |-  `' ( { A }  +c  { B } )  =  ( ( { (/) }  X.  { A }
)  u.  ( { 1o }  X.  { B } ) )
21fveq1i 5857 . . 3  |-  ( `' ( { A }  +c  { B } ) `
 1o )  =  ( ( ( {
(/) }  X.  { A } )  u.  ( { 1o }  X.  { B } ) ) `  1o )
3 vex 3098 . . . . . . . . . . . . 13  |-  x  e. 
_V
4 fvi 5915 . . . . . . . . . . . . 13  |-  ( x  e.  _V  ->  (  _I  `  x )  =  x )
53, 4ax-mp 5 . . . . . . . . . . . 12  |-  (  _I 
`  x )  =  x
6 elsni 4039 . . . . . . . . . . . . 13  |-  ( x  e.  { A }  ->  x  =  A )
76fveq2d 5860 . . . . . . . . . . . 12  |-  ( x  e.  { A }  ->  (  _I  `  x
)  =  (  _I 
`  A ) )
85, 7syl5eqr 2498 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  x  =  (  _I 
`  A ) )
9 elsn 4028 . . . . . . . . . . 11  |-  ( x  e.  { (  _I 
`  A ) }  <-> 
x  =  (  _I 
`  A ) )
108, 9sylibr 212 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  x  e.  { (  _I  `  A ) } )
1110ssriv 3493 . . . . . . . . 9  |-  { A }  C_  { (  _I 
`  A ) }
12 xpss2 5102 . . . . . . . . 9  |-  ( { A }  C_  { (  _I  `  A ) }  ->  ( { (/)
}  X.  { A } )  C_  ( { (/) }  X.  {
(  _I  `  A
) } ) )
1311, 12ax-mp 5 . . . . . . . 8  |-  ( {
(/) }  X.  { A } )  C_  ( { (/) }  X.  {
(  _I  `  A
) } )
14 0ex 4567 . . . . . . . . 9  |-  (/)  e.  _V
15 fvex 5866 . . . . . . . . 9  |-  (  _I 
`  A )  e. 
_V
1614, 15xpsn 6058 . . . . . . . 8  |-  ( {
(/) }  X.  { (  _I  `  A ) } )  =  { <.
(/) ,  (  _I  `  A ) >. }
1713, 16sseqtri 3521 . . . . . . 7  |-  ( {
(/) }  X.  { A } )  C_  { <. (/)
,  (  _I  `  A ) >. }
1814, 15funsn 5626 . . . . . . 7  |-  Fun  { <.
(/) ,  (  _I  `  A ) >. }
19 funss 5596 . . . . . . 7  |-  ( ( { (/) }  X.  { A } )  C_  { <. (/)
,  (  _I  `  A ) >. }  ->  ( Fun  { <. (/) ,  (  _I  `  A )
>. }  ->  Fun  ( {
(/) }  X.  { A } ) ) )
2017, 18, 19mp2 9 . . . . . 6  |-  Fun  ( { (/) }  X.  { A } )
21 funfn 5607 . . . . . 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 5763 . . . 4  |-  ( B  e.  V  ->  ( { 1o }  X.  { B } )  Fn  { 1o } )
25 dmxpss 5428 . . . . . . 7  |-  dom  ( { (/) }  X.  { A } )  C_  { (/) }
26 ssrin 3708 . . . . . . 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 7147 . . . . . . . 8  |-  1o  =/=  (/)
2928necomi 2713 . . . . . . 7  |-  (/)  =/=  1o
30 disjsn2 4076 . . . . . . 7  |-  ( (/)  =/=  1o  ->  ( { (/)
}  i^i  { 1o } )  =  (/) )
3129, 30ax-mp 5 . . . . . 6  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
32 sseq0 3803 . . . . . 6  |-  ( ( ( dom  ( {
(/) }  X.  { A } )  i^i  { 1o } )  C_  ( { (/) }  i^i  { 1o } )  /\  ( { (/) }  i^i  { 1o } )  =  (/) )  ->  ( dom  ( { (/) }  X.  { A } )  i^i  { 1o } )  =  (/) )
3327, 31, 32mp2an 672 . . . . 5  |-  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } )  =  (/)
3433a1i 11 . . . 4  |-  ( B  e.  V  ->  ( dom  ( { (/) }  X.  { A } )  i^i 
{ 1o } )  =  (/) )
35 1on 7139 . . . . . . 7  |-  1o  e.  On
3635elexi 3105 . . . . . 6  |-  1o  e.  _V
3736snid 4042 . . . . 5  |-  1o  e.  { 1o }
3837a1i 11 . . . 4  |-  ( B  e.  V  ->  1o  e.  { 1o } )
39 fvun2 5930 . . . 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 1233 . . 3  |-  ( B  e.  V  ->  (
( ( { (/) }  X.  { A }
)  u.  ( { 1o }  X.  { B } ) ) `  1o )  =  (
( { 1o }  X.  { B } ) `
 1o ) )
412, 40syl5eq 2496 . 2  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  ( ( { 1o }  X.  { B } ) `  1o ) )
42 xpsng 6057 . . . . 5  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( { 1o }  X.  { B } )  =  { <. 1o ,  B >. } )
4342fveq1d 5858 . . . 4  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( ( { 1o }  X.  { B }
) `  1o )  =  ( { <. 1o ,  B >. } `  1o ) )
44 fvsng 6090 . . . 4  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( { <. 1o ,  B >. } `  1o )  =  B )
4543, 44eqtrd 2484 . . 3  |-  ( ( 1o  e.  On  /\  B  e.  V )  ->  ( ( { 1o }  X.  { B }
) `  1o )  =  B )
4635, 45mpan 670 . 2  |-  ( B  e.  V  ->  (
( { 1o }  X.  { B } ) `
 1o )  =  B )
4741, 46eqtrd 2484 1  |-  ( B  e.  V  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020    _I cid 4780   Oncon0 4868    X. cxp 4987   `'ccnv 4988   dom cdm 4989   Fun wfun 5572    Fn wfn 5573   ` cfv 5578  (class class class)co 6281   1oc1o 7125    +c ccda 8550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1o 7132  df-cda 8551
This theorem is referenced by:  xpscfv  14941  xpsfeq  14943  xpsfrnel2  14944  xpsff1o  14947  xpsle  14960  dmdprdpr  17077  dprdpr  17078  xpstopnlem1  20288  xpstopnlem2  20290  xpsxmetlem  20860  xpsdsval  20862  xpsmet  20863
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