Theorem 0rest 17474

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 5307	. . . 4 ⊢ ∅ ∈ V
2		restval 17471	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
3	1, 2	mpan 690	. . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
4		mpt0 6710	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
5	4	rneqi 5948	. . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅
6		rn0 5936	. . . 4 ⊢ ran ∅ = ∅
7	5, 6	eqtri 2765	. . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
8	3, 7	eqtrdi 2793	. 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9		relxp 5703	. . . 4 ⊢ Rel (V × V)
10		restfn 17469	. . . . . 6 ⊢ ↾t Fn (V × V)
11	10	fndmi 6672	. . . . 5 ⊢ dom ↾t = (V × V)
12	11	releqi 5787	. . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V))
13	9, 12	mpbir 231	. . 3 ⊢ Rel dom ↾t
14	13	ovprc2 7471	. 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
15	8, 14	pm2.61i 182	1 ⊢ (∅ ↾t 𝐴) = ∅

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950  ∅c0 4333   ↦ cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690  (class class class)co 7431   ↾_t crest 17465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-rest 17467

This theorem is referenced by:  firest  17477  topnval  17479  resstopn  23194  ussval  24268  bj-rest00  37082