Theorem bj-rest10 33041

Description: An elementwise intersection on a nonempty family by the empty set is the singleton on the empty set. TODO: this generalizes rest0 20973 and could replace it. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-rest10	|- ( X e. V -> ( X =/= (/) -> ( X ↾t (/) ) = { (/) } ) )

Proof of Theorem bj-rest10

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4790	. . . . . 6 |- (/) e. _V
2		elrest 16088	. . . . . 6 |- ( ( X e. V /\ (/) e. _V ) -> ( x e. ( X ↾t (/) ) <-> E. y e. X x = ( y i^i (/) ) ) )
3	1, 2	mpan2 707	. . . . 5 |- ( X e. V -> ( x e. ( X ↾t (/) ) <-> E. y e. X x = ( y i^i (/) ) ) )
4		in0 3968	. . . . . . . . 9 |- ( y i^i (/) ) = (/)
5	4	eqeq2i 2634	. . . . . . . 8 |- ( x = ( y i^i (/) ) <-> x = (/) )
6	5	rexbii 3041	. . . . . . 7 |- ( E. y e. X x = ( y i^i (/) ) <-> E. y e. X x = (/) )
7		df-rex 2918	. . . . . . . 8 |- ( E. y e. X x = (/) <-> E. y ( y e. X /\ x = (/) ) )
8		19.41v 1914	. . . . . . . . 9 |- ( E. y ( y e. X /\ x = (/) ) <-> ( E. y y e. X /\ x = (/) ) )
9		n0 3931	. . . . . . . . . . 11 |- ( X =/= (/) <-> E. y y e. X )
10	9	bicomi 214	. . . . . . . . . 10 |- ( E. y y e. X <-> X =/= (/) )
11	10	anbi1i 731	. . . . . . . . 9 |- ( ( E. y y e. X /\ x = (/) ) <-> ( X =/= (/) /\ x = (/) ) )
12	8, 11	bitri 264	. . . . . . . 8 |- ( E. y ( y e. X /\ x = (/) ) <-> ( X =/= (/) /\ x = (/) ) )
13	7, 12	bitri 264	. . . . . . 7 |- ( E. y e. X x = (/) <-> ( X =/= (/) /\ x = (/) ) )
14	6, 13	bitri 264	. . . . . 6 |- ( E. y e. X x = ( y i^i (/) ) <-> ( X =/= (/) /\ x = (/) ) )
15	14	baib 944	. . . . 5 |- ( X =/= (/) -> ( E. y e. X x = ( y i^i (/) ) <-> x = (/) ) )
16	3, 15	sylan9bb 736	. . . 4 |- ( ( X e. V /\ X =/= (/) ) -> ( x e. ( X ↾t (/) ) <-> x = (/) ) )
17		velsn 4193	. . . 4 |- ( x e. { (/) } <-> x = (/) )
18	16, 17	syl6bbr 278	. . 3 |- ( ( X e. V /\ X =/= (/) ) -> ( x e. ( X ↾t (/) ) <-> x e. { (/) } ) )
19	18	eqrdv 2620	. 2 |- ( ( X e. V /\ X =/= (/) ) -> ( X ↾t (/) ) = { (/) } )
20	19	ex 450	1 |- ( X e. V -> ( X =/= (/) -> ( X ↾t (/) ) = { (/) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   (/)c0 3915   {csn 4177  (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-rest10b  33042