Theorem iinvdif 4592

Description: The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)

Assertion

Ref	Expression
iinvdif	|- |^|_ x e. A ( _V \ B ) = ( _V \ U_ x e. A B )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem iinvdif

Step	Hyp	Ref	Expression
1		dif0 3950	. . . 4 |- ( _V \ (/) ) = _V
2		0iun 4577	. . . . 5 |- U_ x e. (/) B = (/)
3	2	difeq2i 3725	. . . 4 |- ( _V \ U_ x e. (/) B ) = ( _V \ (/) )
4		0iin 4578	. . . 4 |- |^|_ x e. (/) ( _V \ B ) = _V
5	1, 3, 4	3eqtr4ri 2655	. . 3 |- |^|_ x e. (/) ( _V \ B ) = ( _V \ U_ x e. (/) B )
6		iineq1 4535	. . 3 |- ( A = (/) -> |^|_ x e. A ( _V \ B ) = |^|_ x e. (/) ( _V \ B ) )
7		iuneq1 4534	. . . 4 |- ( A = (/) -> U_ x e. A B = U_ x e. (/) B )
8	7	difeq2d 3728	. . 3 |- ( A = (/) -> ( _V \ U_ x e. A B ) = ( _V \ U_ x e. (/) B ) )
9	5, 6, 8	3eqtr4a 2682	. 2 |- ( A = (/) -> |^|_ x e. A ( _V \ B ) = ( _V \ U_ x e. A B ) )
10		iindif2 4589	. 2 |- ( A =/= (/) -> |^|_ x e. A ( _V \ B ) = ( _V \ U_ x e. A B ) )
11	9, 10	pm2.61ine 2877	1 |- |^|_ x e. A ( _V \ B ) = ( _V \ U_ x e. A B )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   (/)c0 3915   U_ciun 4520   |^|_ciin 4521

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iun 4522  df-iin 4523

This theorem is referenced by: (None)