Theorem bj-restb 33047

Description: An elementwise intersection by a set on a family containing a superset of that set contains that set. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restb	|- ( X e. V -> ( ( A C_ B /\ B e. X ) -> A e. ( X ↾t A ) ) )

Proof of Theorem bj-restb

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . 8 |- ( A C_ B -> A C_ B )
2		ssid 3624	. . . . . . . . 9 |- A C_ A
3	2	a1i 11	. . . . . . . 8 |- ( A C_ B -> A C_ A )
4	1, 3	ssind 3837	. . . . . . 7 |- ( A C_ B -> A C_ ( B i^i A ) )
5		inss2 3834	. . . . . . . 8 |- ( B i^i A ) C_ A
6	5	a1i 11	. . . . . . 7 |- ( A C_ B -> ( B i^i A ) C_ A )
7	4, 6	eqssd 3620	. . . . . 6 |- ( A C_ B -> A = ( B i^i A ) )
8		eleq1 2689	. . . . . . . . . 10 |- ( y = B -> ( y e. X <-> B e. X ) )
9		ineq1 3807	. . . . . . . . . . 11 |- ( y = B -> ( y i^i A ) = ( B i^i A ) )
10	9	eqeq2d 2632	. . . . . . . . . 10 |- ( y = B -> ( A = ( y i^i A ) <-> A = ( B i^i A ) ) )
11	8, 10	anbi12d 747	. . . . . . . . 9 |- ( y = B -> ( ( y e. X /\ A = ( y i^i A ) ) <-> ( B e. X /\ A = ( B i^i A ) ) ) )
12	11	spcegv 3294	. . . . . . . 8 |- ( B e. X -> ( ( B e. X /\ A = ( B i^i A ) ) -> E. y ( y e. X /\ A = ( y i^i A ) ) ) )
13	12	expd 452	. . . . . . 7 |- ( B e. X -> ( B e. X -> ( A = ( B i^i A ) -> E. y ( y e. X /\ A = ( y i^i A ) ) ) ) )
14	13	pm2.43i 52	. . . . . 6 |- ( B e. X -> ( A = ( B i^i A ) -> E. y ( y e. X /\ A = ( y i^i A ) ) ) )
15	7, 14	mpan9 486	. . . . 5 |- ( ( A C_ B /\ B e. X ) -> E. y ( y e. X /\ A = ( y i^i A ) ) )
16		df-rex 2918	. . . . 5 |- ( E. y e. X A = ( y i^i A ) <-> E. y ( y e. X /\ A = ( y i^i A ) ) )
17	15, 16	sylibr 224	. . . 4 |- ( ( A C_ B /\ B e. X ) -> E. y e. X A = ( y i^i A ) )
18	17	adantl 482	. . 3 |- ( ( X e. V /\ ( A C_ B /\ B e. X ) ) -> E. y e. X A = ( y i^i A ) )
19		ssexg 4804	. . . 4 |- ( ( A C_ B /\ B e. X ) -> A e. _V )
20		elrest 16088	. . . 4 |- ( ( X e. V /\ A e. _V ) -> ( A e. ( X ↾t A ) <-> E. y e. X A = ( y i^i A ) ) )
21	19, 20	sylan2 491	. . 3 |- ( ( X e. V /\ ( A C_ B /\ B e. X ) ) -> ( A e. ( X ↾t A ) <-> E. y e. X A = ( y i^i A ) ) )
22	18, 21	mpbird 247	. 2 |- ( ( X e. V /\ ( A C_ B /\ B e. X ) ) -> A e. ( X ↾t A ) )
23	22	ex 450	1 |- ( X e. V -> ( ( A C_ B /\ B e. X ) -> A e. ( X ↾t A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574  (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:  bj-restv  33048  bj-resta  33049