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Theorem xpscfv 14492
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 3892 . . . 4  |-  ( C  e.  { (/) ,  1o }  ->  ( C  =  (/)  \/  C  =  1o ) )
2 df2o3 6925 . . . 4  |-  2o  =  { (/) ,  1o }
31, 2eleq2s 2530 . . 3  |-  ( C  e.  2o  ->  ( C  =  (/)  \/  C  =  1o ) )
4 xpsc0 14490 . . . . . 6  |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
54adantr 465 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
6 fveq2 5686 . . . . . 6  |-  ( C  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 C )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
7 iftrue 3792 . . . . . 6  |-  ( C  =  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  A )
86, 7eqeq12d 2452 . . . . 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 14491 . . . . . 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 5686 . . . . . 6  |-  ( C  =  1o  ->  ( `' ( { A }  +c  { B }
) `  C )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
13 1n0 6927 . . . . . . . 8  |-  1o  =/=  (/)
14 neeq1 2611 . . . . . . . 8  |-  ( C  =  1o  ->  ( C  =/=  (/)  <->  1o  =/=  (/) ) )
1513, 14mpbiri 233 . . . . . . 7  |-  ( C  =  1o  ->  C  =/=  (/) )
16 ifnefalse 3796 . . . . . . 7  |-  ( C  =/=  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1715, 16syl 16 . . . . . 6  |-  ( C  =  1o  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1812, 17eqeq12d 2452 . . . . 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 1184 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 1369    e. wcel 1756    =/= wne 2601   (/)c0 3632   ifcif 3786   {csn 3872   {cpr 3874   `'ccnv 4834   ` cfv 5413  (class class class)co 6086   1oc1o 6905   2oc2o 6906    +c ccda 8328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1o 6912  df-2o 6913  df-cda 8329
This theorem is referenced by:  xpsfrn2  14500  xpslem  14503  xpsaddlem  14505  xpsvsca  14509
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