Theorem bj-restuni2 33051

Description: The union of an elementwise intersection on a family of sets by a subset is equal to that subset. See also restuni 20966 and restuni2 20971. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restuni2	|- ( ( X e. V /\ A C_ U. X ) -> U. ( X ↾t A ) = A )

Proof of Theorem bj-restuni2

Step	Hyp	Ref	Expression
1		uniexg 6955	. . . . 5 |- ( X e. V -> U. X e. _V )
2		ssexg 4804	. . . . 5 |- ( ( A C_ U. X /\ U. X e. _V ) -> A e. _V )
3	1, 2	sylan2 491	. . . 4 |- ( ( A C_ U. X /\ X e. V ) -> A e. _V )
4	3	ancoms 469	. . 3 |- ( ( X e. V /\ A C_ U. X ) -> A e. _V )
5		bj-restuni 33050	. . 3 |- ( ( X e. V /\ A e. _V ) -> U. ( X ↾t A ) = ( U. X i^i A ) )
6	4, 5	syldan 487	. 2 |- ( ( X e. V /\ A C_ U. X ) -> U. ( X ↾t A ) = ( U. X i^i A ) )
7		inss2 3834	. . . . 5 |- ( U. X i^i A ) C_ A
8	7	a1i 11	. . . 4 |- ( A C_ U. X -> ( U. X i^i A ) C_ A )
9		id 22	. . . . 5 |- ( A C_ U. X -> A C_ U. X )
10		ssid 3624	. . . . . 6 |- A C_ A
11	10	a1i 11	. . . . 5 |- ( A C_ U. X -> A C_ A )
12	9, 11	ssind 3837	. . . 4 |- ( A C_ U. X -> A C_ ( U. X i^i A ) )
13	8, 12	eqssd 3620	. . 3 |- ( A C_ U. X -> ( U. X i^i A ) = A )
14	13	adantl 482	. 2 |- ( ( X e. V /\ A C_ U. X ) -> ( U. X i^i A ) = A )
15	6, 14	eqtrd 2656	1 |- ( ( X e. V /\ A C_ U. X ) -> U. ( X ↾t A ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436  (class class class)co 6650   ↾_t crest 16081

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083

This theorem is referenced by: (None)