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Theorem xpscfv 14831
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 4030 . . . 4  |-  ( C  e.  { (/) ,  1o }  ->  ( C  =  (/)  \/  C  =  1o ) )
2 df2o3 7141 . . . 4  |-  2o  =  { (/) ,  1o }
31, 2eleq2s 2549 . . 3  |-  ( C  e.  2o  ->  ( C  =  (/)  \/  C  =  1o ) )
4 xpsc0 14829 . . . . . 6  |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
54adantr 465 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
6 fveq2 5852 . . . . . 6  |-  ( C  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 C )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
7 iftrue 3928 . . . . . 6  |-  ( C  =  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  A )
86, 7eqeq12d 2463 . . . . 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 14830 . . . . . 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 5852 . . . . . 6  |-  ( C  =  1o  ->  ( `' ( { A }  +c  { B }
) `  C )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
13 1n0 7143 . . . . . . . 8  |-  1o  =/=  (/)
14 neeq1 2722 . . . . . . . 8  |-  ( C  =  1o  ->  ( C  =/=  (/)  <->  1o  =/=  (/) ) )
1513, 14mpbiri 233 . . . . . . 7  |-  ( C  =  1o  ->  C  =/=  (/) )
16 ifnefalse 3934 . . . . . . 7  |-  ( C  =/=  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1715, 16syl 16 . . . . . 6  |-  ( C  =  1o  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1812, 17eqeq12d 2463 . . . . 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 1192 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   (/)c0 3767   ifcif 3922   {csn 4010   {cpr 4012   `'ccnv 4984   ` cfv 5574  (class class class)co 6277   1oc1o 7121   2oc2o 7122    +c ccda 8545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1o 7128  df-2o 7129  df-cda 8546
This theorem is referenced by:  xpsfrn2  14839  xpslem  14842  xpsaddlem  14844  xpsvsca  14848
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