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Theorem bj-projeq 31115
Description: Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-projeq  |-  ( A  =  C  ->  ( B  =  D  ->  ( A Proj  B )  =  ( C Proj  D ) ) )

Proof of Theorem bj-projeq
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 459 . . . . . 6  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
2 simpl 455 . . . . . . 7  |-  ( ( A  =  C  /\  B  =  D )  ->  A  =  C )
32sneqd 3984 . . . . . 6  |-  ( ( A  =  C  /\  B  =  D )  ->  { A }  =  { C } )
41, 3imaeq12d 5158 . . . . 5  |-  ( ( A  =  C  /\  B  =  D )  ->  ( B " { A } )  =  ( D " { C } ) )
54eleq2d 2472 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( { x }  e.  ( B " { A } )  <->  { x }  e.  ( D " { C } ) ) )
65abbidv 2538 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  { x  |  {
x }  e.  ( B " { A } ) }  =  { x  |  {
x }  e.  ( D " { C } ) } )
7 df-bj-proj 31114 . . 3  |-  ( A Proj 
B )  =  {
x  |  { x }  e.  ( B " { A } ) }
8 df-bj-proj 31114 . . 3  |-  ( C Proj 
D )  =  {
x  |  { x }  e.  ( D " { C } ) }
96, 7, 83eqtr4g 2468 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A Proj  B )  =  ( C Proj  D
) )
109ex 432 1  |-  ( A  =  C  ->  ( B  =  D  ->  ( A Proj  B )  =  ( C Proj  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   {csn 3972   "cima 4826   Proj bj-cproj 31113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-bj-proj 31114
This theorem is referenced by:  bj-projeq2  31116
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