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Theorem xpscfv 14969
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 3964 . . . 4  |-  ( C  e.  { (/) ,  1o }  ->  ( C  =  (/)  \/  C  =  1o ) )
2 df2o3 7061 . . . 4  |-  2o  =  { (/) ,  1o }
31, 2eleq2s 2490 . . 3  |-  ( C  e.  2o  ->  ( C  =  (/)  \/  C  =  1o ) )
4 xpsc0 14967 . . . . . 6  |-  ( A  e.  V  ->  ( `' ( { A }  +c  { B }
) `  (/) )  =  A )
54adantr 463 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( `' ( { A }  +c  { B } ) `  (/) )  =  A )
6 fveq2 5774 . . . . . 6  |-  ( C  =  (/)  ->  ( `' ( { A }  +c  { B } ) `
 C )  =  ( `' ( { A }  +c  { B } ) `  (/) ) )
7 iftrue 3863 . . . . . 6  |-  ( C  =  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  A )
86, 7eqeq12d 2404 . . . . 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 14968 . . . . . 6  |-  ( B  e.  W  ->  ( `' ( { A }  +c  { B }
) `  1o )  =  B )
1110adantl 464 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( `' ( { A }  +c  { B } ) `  1o )  =  B )
12 fveq2 5774 . . . . . 6  |-  ( C  =  1o  ->  ( `' ( { A }  +c  { B }
) `  C )  =  ( `' ( { A }  +c  { B } ) `  1o ) )
13 1n0 7063 . . . . . . . 8  |-  1o  =/=  (/)
14 neeq1 2663 . . . . . . . 8  |-  ( C  =  1o  ->  ( C  =/=  (/)  <->  1o  =/=  (/) ) )
1513, 14mpbiri 233 . . . . . . 7  |-  ( C  =  1o  ->  C  =/=  (/) )
16 ifnefalse 3869 . . . . . . 7  |-  ( C  =/=  (/)  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1715, 16syl 16 . . . . . 6  |-  ( C  =  1o  ->  if ( C  =  (/) ,  A ,  B )  =  B )
1812, 17eqeq12d 2404 . . . . 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 378 . . 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 1191 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 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   (/)c0 3711   ifcif 3857   {csn 3944   {cpr 3946   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   1oc1o 7041   2oc2o 7042    +c ccda 8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1o 7048  df-2o 7049  df-cda 8461
This theorem is referenced by:  xpsfrn2  14977  xpslem  14980  xpsaddlem  14982  xpsvsca  14986
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