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Theorem bj-xpimasn 33470
Description: The image of a singleton, general case. [Change and relabel xpimasn 5445 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-xpimasn  |-  ( ( A  X.  B )
" { X }
)  =  if ( X  e.  A ,  B ,  (/) )

Proof of Theorem bj-xpimasn
StepHypRef Expression
1 xpima 5442 . 2  |-  ( ( A  X.  B )
" { X }
)  =  if ( ( A  i^i  { X } )  =  (/) ,  (/) ,  B )
2 disjsn 4083 . . 3  |-  ( ( A  i^i  { X } )  =  (/)  <->  -.  X  e.  A )
3 eqid 2462 . . 3  |-  B  =  B
42, 3ifbieq2i 3958 . 2  |-  if ( ( A  i^i  { X } )  =  (/) ,  (/) ,  B )  =  if ( -.  X  e.  A ,  (/) ,  B
)
5 ifnot 3979 . 2  |-  if ( -.  X  e.  A ,  (/) ,  B )  =  if ( X  e.  A ,  B ,  (/) )
61, 4, 53eqtri 2495 1  |-  ( ( A  X.  B )
" { X }
)  =  if ( X  e.  A ,  B ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1374    e. wcel 1762    i^i cin 3470   (/)c0 3780   ifcif 3934   {csn 4022    X. cxp 4992   "cima 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007
This theorem is referenced by:  bj-xpima1sn  33471  bj-xpima2sn  33473
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