Theorem invdisjrab 4053

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 2350	. . . . 5 |- F/_ x z
2		nfcv 2350	. . . . 5 |- F/_ x B
3		nfcsb1v 3134	. . . . . 6 |- F/_ x [_ z / x ]_ C
4	3	nfeq1 2360	. . . . 5 |- F/ x [_ z / x ]_ C = y
5		csbeq1a 3110	. . . . . 6 |- ( x = z -> C = [_ z / x ]_ C )
6	5	eqeq1d 2216	. . . . 5 |- ( x = z -> ( C = y <-> [_ z / x ]_ C = y ) )
7	1, 2, 4, 6	elrabf 2934	. . . 4 |- ( z e. { x e. B | C = y } <-> ( z e. B /\ [_ z / x ]_ C = y ) )
8		simprr 531	. . . 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 2594	. 2 |- A. y e. A A. z e. { x e. B | C = y } [_ z / x ]_ C = y
11		invdisj 4052	. 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 1373   e. wcel 2178   A.wral 2486   {crab 2490   [_csb 3101  Disj wdisj 4035

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-disj 4036

This theorem is referenced by:  disjwrdpfx  11191