Theorem bj-restsnid 33040

Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 33035 and bj-restsnss 33036. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restsnid	|- ( { A } ↾t A ) = { A }

Proof of Theorem bj-restsnid

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

Step	Hyp	Ref	Expression
1		ssid 3624	. . 3 |- A C_ A
2		bj-restsnss 33036	. . 3 |- ( ( A e. _V /\ A C_ A ) -> ( { A } ↾t A ) = { A } )
3	1, 2	mpan2 707	. 2 |- ( A e. _V -> ( { A } ↾t A ) = { A } )
4		df-rest 16083	. . . . 5 |- ↾t = ( x e. _V , y e. _V |-> ran ( z e. x |-> ( z i^i y ) ) )
5	4	reldmmpt2 6771	. . . 4 |- Rel dom ↾t
6	5	ovprc2 6685	. . 3 |- ( -. A e. _V -> ( { A } ↾t A ) = (/) )
7		snprc 4253	. . . 4 |- ( -. A e. _V <-> { A } = (/) )
8	7	biimpi 206	. . 3 |- ( -. A e. _V -> { A } = (/) )
9	6, 8	eqtr4d 2659	. 2 |- ( -. A e. _V -> ( { A } ↾t A ) = { A } )
10	3, 9	pm2.61i 176	1 |- ( { A } ↾t A ) = { A }

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   ran crn 5115  (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-pow 4843  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)