Theorem bj-restsn 33035

Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 33038 and bj-restsnid 33040. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restsn	|- ( ( Y e. V /\ A e. W ) -> ( { Y } ↾t A ) = { ( Y i^i A ) } )

Proof of Theorem bj-restsn

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

Step	Hyp	Ref	Expression
1		snex 4908	. . . 4 |- { Y } e. _V
2		elrest 16088	. . . 4 |- ( ( { Y } e. _V /\ A e. W ) -> ( x e. ( { Y } ↾t A ) <-> E. y e. { Y } x = ( y i^i A ) ) )
3	1, 2	mpan 706	. . 3 |- ( A e. W -> ( x e. ( { Y } ↾t A ) <-> E. y e. { Y } x = ( y i^i A ) ) )
4		velsn 4193	. . . . 5 |- ( x e. { ( y i^i A ) } <-> x = ( y i^i A ) )
5		ineq1 3807	. . . . . . 7 |- ( y = Y -> ( y i^i A ) = ( Y i^i A ) )
6	5	sneqd 4189	. . . . . 6 |- ( y = Y -> { ( y i^i A ) } = { ( Y i^i A ) } )
7	6	eleq2d 2687	. . . . 5 |- ( y = Y -> ( x e. { ( y i^i A ) } <-> x e. { ( Y i^i A ) } ) )
8	4, 7	syl5bbr 274	. . . 4 |- ( y = Y -> ( x = ( y i^i A ) <-> x e. { ( Y i^i A ) } ) )
9	8	rexsng 4219	. . 3 |- ( Y e. V -> ( E. y e. { Y } x = ( y i^i A ) <-> x e. { ( Y i^i A ) } ) )
10	3, 9	sylan9bbr 737	. 2 |- ( ( Y e. V /\ A e. W ) -> ( x e. ( { Y } ↾t A ) <-> x e. { ( Y i^i A ) } ) )
11	10	eqrdv 2620	1 |- ( ( Y e. V /\ A e. W ) -> ( { Y } ↾t A ) = { ( Y i^i A ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   {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-restsnss  33036  bj-restsnss2  33037