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Theorem xpscfv 14618
Description: The value of the pair function at an element of  2o. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
xpscfv  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  C
)  =  if ( C  =  (/) ,  A ,  B ) )

Proof of Theorem xpscfv
StepHypRef Expression
1 elpri 4004 . . . 4  |-  ( C  e.  { (/) ,  1o }  ->  ( C  =  (/)  \/  C  =  1o ) )
2 df2o3 7042 . . . 4  |-  2o  =  { (/) ,  1o }
31, 2eleq2s 2562 . . 3  |-  ( C  e.  2o  ->  ( C  =  (/)  \/  C  =  1o ) )
4 xpsc0 14616 . . . . . 6  |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
54adantr 465 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
6 fveq2 5798 . . . . . 6  |-  ( C  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 C )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
7 iftrue 3904 . . . . . 6  |-  ( C  =  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  A )
86, 7eqeq12d 2476 . . . . 5  |-  ( C  =  (/)  ->  ( ( `' ( { A }  +c  { B }
) `  C )  =  if ( C  =  (/) ,  A ,  B
)  <->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A ) )
95, 8syl5ibrcom 222 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  =  (/)  ->  ( `' ( { A }  +c  { B } ) `  C
)  =  if ( C  =  (/) ,  A ,  B ) ) )
10 xpsc1 14617 . . . . . 6  |-  ( B  e.  W  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
1110adantl 466 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
12 fveq2 5798 . . . . . 6  |-  ( C  =  1o  ->  ( `' ( { A }  +c  { B }
) `  C )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
13 1n0 7044 . . . . . . . 8  |-  1o  =/=  (/)
14 neeq1 2732 . . . . . . . 8  |-  ( C  =  1o  ->  ( C  =/=  (/)  <->  1o  =/=  (/) ) )
1513, 14mpbiri 233 . . . . . . 7  |-  ( C  =  1o  ->  C  =/=  (/) )
16 ifnefalse 3908 . . . . . . 7  |-  ( C  =/=  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1715, 16syl 16 . . . . . 6  |-  ( C  =  1o  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1812, 17eqeq12d 2476 . . . . 5  |-  ( C  =  1o  ->  (
( `' ( { A }  +c  { B } ) `  C
)  =  if ( C  =  (/) ,  A ,  B )  <->  ( `' ( { A }  +c  { B } ) `  1o )  =  B
) )
1911, 18syl5ibrcom 222 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  =  1o 
->  ( `' ( { A }  +c  { B } ) `  C
)  =  if ( C  =  (/) ,  A ,  B ) ) )
209, 19jaod 380 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( C  =  (/)  \/  C  =  1o )  ->  ( `' ( { A }  +c  { B } ) `  C )  =  if ( C  =  (/) ,  A ,  B ) ) )
213, 20syl5 32 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  2o  ->  ( `' ( { A }  +c  { B } ) `  C
)  =  if ( C  =  (/) ,  A ,  B ) ) )
22213impia 1185 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  2o )  ->  ( `' ( { A }  +c  { B } ) `  C
)  =  if ( C  =  (/) ,  A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   (/)c0 3744   ifcif 3898   {csn 3984   {cpr 3986   `'ccnv 4946   ` cfv 5525  (class class class)co 6199   1oc1o 7022   2oc2o 7023    +c ccda 8446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1o 7029  df-2o 7030  df-cda 8447
This theorem is referenced by:  xpsfrn2  14626  xpslem  14629  xpsaddlem  14631  xpsvsca  14635
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