Theorem 0rest 16536

Description: Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
0rest	⊢ (∅ ↾t 𝐴) = ∅

Proof of Theorem 0rest

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5109	. . . 4 ⊢ ∅ ∈ V
2		restval 16533	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
3	1, 2	mpan 686	. . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
4		mpt0 6365	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
5	4	rneqi 5696	. . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅
6		rn0 5722	. . . 4 ⊢ ran ∅ = ∅
7	5, 6	eqtri 2821	. . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
8	3, 7	syl6eq 2849	. 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9		relxp 5468	. . . 4 ⊢ Rel (V × V)
10		restfn 16531	. . . . . 6 ⊢ ↾t Fn (V × V)
11		fndm 6332	. . . . . 6 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
12	10, 11	ax-mp 5	. . . . 5 ⊢ dom ↾t = (V × V)
13	12	releqi 5545	. . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V))
14	9, 13	mpbir 232	. . 3 ⊢ Rel dom ↾t
15	14	ovprc2 7062	. 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
16	8, 15	pm2.61i 183	1 ⊢ (∅ ↾t 𝐴) = ∅

Syntax hints:   = wceq 1525   ∈ wcel 2083  Vcvv 3440   ∩ cin 3864  ∅c0 4217   ↦ cmpt 5047   × cxp 5448  dom cdm 5450  ran crn 5451  Rel wrel 5455   Fn wfn 6227  (class class class)co 7023   ↾_t crest 16527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-rest 16529

This theorem is referenced by:  firest  16539  topnval  16541  resstopn  21482  ussval  22555  bj-rest00  33992