Theorem 0rest 17121

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 5234	. . . 4 ⊢ ∅ ∈ V
2		restval 17118	. . . 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 6571	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
5	4	rneqi 5843	. . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅
6		rn0 5832	. . . 4 ⊢ ran ∅ = ∅
7	5, 6	eqtri 2767	. . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
8	3, 7	eqtrdi 2795	. 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9		relxp 5606	. . . 4 ⊢ Rel (V × V)
10		restfn 17116	. . . . . 6 ⊢ ↾t Fn (V × V)
11	10	fndmi 6533	. . . . 5 ⊢ dom ↾t = (V × V)
12	11	releqi 5686	. . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V))
13	9, 12	mpbir 230	. . 3 ⊢ Rel dom ↾t
14	13	ovprc2 7308	. 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
15	8, 14	pm2.61i 182	1 ⊢ (∅ ↾t 𝐴) = ∅

Syntax hints:   = wceq 1541   ∈ wcel 2109  Vcvv 3430   ∩ cin 3890  ∅c0 4261   ↦ cmpt 5161   × cxp 5586  dom cdm 5588  ran crn 5589  Rel wrel 5593  (class class class)co 7268   ↾_t crest 17112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-rest 17114

This theorem is referenced by:  firest  17124  topnval  17126  resstopn  22318  ussval  23392  bj-rest00  35231