Theorem invdisjrab 4102

Description: The restricted class abstractions { x e. B | C = y } for distinct y e. A are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)

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 2384	. . . . 5 |- F/_ x z
2		nfcv 2384	. . . . 5 |- F/_ x B
3		nfcsb1v 3170	. . . . . 6 |- F/_ x [_ z / x ]_ C
4	3	nfeq1 2394	. . . . 5 |- F/ x [_ z / x ]_ C = y
5		csbeq1a 3146	. . . . . 6 |- ( x = z -> C = [_ z / x ]_ C )
6	5	eqeq1d 2241	. . . . 5 |- ( x = z -> ( C = y <-> [_ z / x ]_ C = y ) )
7	1, 2, 4, 6	elrabf 2970	. . . 4 |- ( z e. { x e. B | C = y } <-> ( z e. B /\ [_ z / x ]_ C = y ) )
8		simprr 533	. . . 4 |- ( ( y e. A /\ ( z e. B /\ [_ z / x ]_ C = y ) ) -> [_ z / x ]_ C = y )
9	7, 8	sylan2b 287	. . 3 |- ( ( y e. A /\ z e. { x e. B | C = y } ) -> [_ z / x ]_ C = y )
10	9	rgen2 2628	. 2 |- A. y e. A A. z e. { x e. B | C = y } [_ z / x ]_ C = y
11		invdisj 4101	. 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 } )
12	10, 11	ax-mp 5	1 |- Disj y e. A { x e. B | C = y }

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2203   A.wral 2520   {crab 2524   [_csb 3137  Disj wdisj 4084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-disj 4085

This theorem is referenced by:  disjwrdpfx  11388