Theorem invdisjrab 4639

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.

Step	Hyp	Ref	Expression
1		nfcv 2764	. . . . . 6 |- F/_ x z
2		nfcv 2764	. . . . . 6 |- F/_ x B
3		nfcsb1v 3549	. . . . . . 7 |- F/_ x [_ z / x ]_ C
4	3	nfeq1 2778	. . . . . 6 |- F/ x [_ z / x ]_ C = y
5		csbeq1a 3542	. . . . . . 7 |- ( x = z -> C = [_ z / x ]_ C )
6	5	eqeq1d 2624	. . . . . 6 |- ( x = z -> ( C = y <-> [_ z / x ]_ C = y ) )
7	1, 2, 4, 6	elrabf 3360	. . . . 5 |- ( z e. { x e. B | C = y } <-> ( z e. B /\ [_ z / x ]_ C = y ) )
8		ax-1 6	. . . . 5 |- ( [_ z / x ]_ C = y -> ( y e. A -> [_ z / x ]_ C = y ) )
9	7, 8	simplbiim 659	. . . 4 |- ( z e. { x e. B | C = y } -> ( y e. A -> [_ z / x ]_ C = y ) )
10	9	impcom 446	. . 3 |- ( ( y e. A /\ z e. { x e. B | C = y } ) -> [_ z / x ]_ C = y )
11	10	rgen2 2975	. 2 |- A. y e. A A. z e. { x e. B | C = y } [_ z / x ]_ C = y
12		invdisj 4638	. 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 } )
13	11, 12	ax-mp 5	1 |- Disj y e. A { x e. B | C = y }

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   [_csb 3533  Disj wdisj 4620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-disj 4621

This theorem is referenced by:  disjxwrd  13455  disjwrdpfx  41408