Theorem bj-rest0 33046

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

Assertion

Ref	Expression
bj-rest0	|- ( ( X e. V /\ A e. W ) -> ( (/) e. X -> (/) e. ( X ↾t A ) ) )

Proof of Theorem bj-rest0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		in0 3968	. . . . 5 |- ( A i^i (/) ) = (/)
2		incom 3805	. . . . 5 |- ( A i^i (/) ) = ( (/) i^i A )
3	1, 2	eqtr3i 2646	. . . 4 |- (/) = ( (/) i^i A )
4		0ex 4790	. . . . 5 |- (/) e. _V
5		eleq1 2689	. . . . . 6 |- ( x = (/) -> ( x e. X <-> (/) e. X ) )
6		ineq1 3807	. . . . . . 7 |- ( x = (/) -> ( x i^i A ) = ( (/) i^i A ) )
7	6	eqeq2d 2632	. . . . . 6 |- ( x = (/) -> ( (/) = ( x i^i A ) <-> (/) = ( (/) i^i A ) ) )
8	5, 7	anbi12d 747	. . . . 5 |- ( x = (/) -> ( ( x e. X /\ (/) = ( x i^i A ) ) <-> ( (/) e. X /\ (/) = ( (/) i^i A ) ) ) )
9	4, 8	spcev 3300	. . . 4 |- ( ( (/) e. X /\ (/) = ( (/) i^i A ) ) -> E. x ( x e. X /\ (/) = ( x i^i A ) ) )
10	3, 9	mpan2 707	. . 3 |- ( (/) e. X -> E. x ( x e. X /\ (/) = ( x i^i A ) ) )
11		df-rex 2918	. . 3 |- ( E. x e. X (/) = ( x i^i A ) <-> E. x ( x e. X /\ (/) = ( x i^i A ) ) )
12	10, 11	sylibr 224	. 2 |- ( (/) e. X -> E. x e. X (/) = ( x i^i A ) )
13		elrest 16088	. 2 |- ( ( X e. V /\ A e. W ) -> ( (/) e. ( X ↾t A ) <-> E. x e. X (/) = ( x i^i A ) ) )
14	12, 13	syl5ibr 236	1 |- ( ( X e. V /\ A e. W ) -> ( (/) e. X -> (/) e. ( X ↾t A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   i^i cin 3573   (/)c0 3915  (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)