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Theorem invdisjrab 4357
Description: The restricted class abstractions  { x  e.  B  |  C  =  y } for distinct  y  e.  A are disjoint. (Contributed by AV, 6-May-2020.)
Assertion
Ref Expression
invdisjrab  |- Disj  y  e.  A  { x  e.  B  |  C  =  y }
Distinct variable groups:    x, B    y, C    x, y
Allowed substitution hints:    A( x, y)    B( y)    C( x)

Proof of Theorem invdisjrab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfcv 2544 . . . . . 6  |-  F/_ x
z
2 nfcv 2544 . . . . . 6  |-  F/_ x B
3 nfcsb1v 3364 . . . . . . 7  |-  F/_ x [_ z  /  x ]_ C
43nfeq1 2559 . . . . . 6  |-  F/ x [_ z  /  x ]_ C  =  y
5 csbeq1a 3357 . . . . . . 7  |-  ( x  =  z  ->  C  =  [_ z  /  x ]_ C )
65eqeq1d 2384 . . . . . 6  |-  ( x  =  z  ->  ( C  =  y  <->  [_ z  /  x ]_ C  =  y ) )
71, 2, 4, 6elrabf 3180 . . . . 5  |-  ( z  e.  { x  e.  B  |  C  =  y }  <->  ( z  e.  B  /\  [_ z  /  x ]_ C  =  y ) )
8 ax-1 6 . . . . . 6  |-  ( [_ z  /  x ]_ C  =  y  ->  ( y  e.  A  ->  [_ z  /  x ]_ C  =  y ) )
98adantl 464 . . . . 5  |-  ( ( z  e.  B  /\  [_ z  /  x ]_ C  =  y )  ->  ( y  e.  A  ->  [_ z  /  x ]_ C  =  y
) )
107, 9sylbi 195 . . . 4  |-  ( z  e.  { x  e.  B  |  C  =  y }  ->  (
y  e.  A  ->  [_ z  /  x ]_ C  =  y
) )
1110impcom 428 . . 3  |-  ( ( y  e.  A  /\  z  e.  { x  e.  B  |  C  =  y } )  ->  [_ z  /  x ]_ C  =  y
)
1211rgen2 2807 . 2  |-  A. y  e.  A  A. z  e.  { x  e.  B  |  C  =  y } [_ z  /  x ]_ C  =  y
13 invdisj 4356 . 2  |-  ( A. y  e.  A  A. z  e.  { x  e.  B  |  C  =  y } [_ z  /  x ]_ C  =  y  -> Disj  y  e.  A  { x  e.  B  |  C  =  y } )
1412, 13ax-mp 5 1  |- Disj  y  e.  A  { x  e.  B  |  C  =  y }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   {crab 2736   [_csb 3348  Disj wdisj 4338
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-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2737  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-disj 4339
This theorem is referenced by:  disjxwrd  12591  disjwrdpfx  32583
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