Theorem 0rest 15913

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 4718	. . . 4 ⊢ ∅ ∈ V
2		restval 15910	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
3	1, 2	mpan 702	. . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
4		mpt0 5934	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
5	4	rneqi 5273	. . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅
6		rn0 5298	. . . 4 ⊢ ran ∅ = ∅
7	5, 6	eqtri 2632	. . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
8	3, 7	syl6eq 2660	. 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9		relxp 5150	. . . 4 ⊢ Rel (V × V)
10		restfn 15908	. . . . . 6 ⊢ ↾t Fn (V × V)
11		fndm 5904	. . . . . 6 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
12	10, 11	ax-mp 5	. . . . 5 ⊢ dom ↾t = (V × V)
13	12	releqi 5125	. . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V))
14	9, 13	mpbir 220	. . 3 ⊢ Rel dom ↾t
15	14	ovprc2 6583	. 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
16	8, 15	pm2.61i 175	1 ⊢ (∅ ↾t 𝐴) = ∅

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  Rel wrel 5043   Fn wfn 5799  (class class class)co 6549   ↾_t crest 15904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906

This theorem is referenced by:  firest  15916  topnval  15918  resstopn  20800  ussval  21873  bj-rest00  32215