Theorem 0rest 16137

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 4823	. . . 4 ⊢ ∅ ∈ V
2		restval 16134	. . . 4 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
3	1, 2	mpan 706	. . 3 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)))
4		mpt0 6059	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
5	4	rneqi 5384	. . . 4 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ran ∅
6		rn0 5409	. . . 4 ⊢ ran ∅ = ∅
7	5, 6	eqtri 2673	. . 3 ⊢ ran (𝑥 ∈ ∅ ↦ (𝑥 ∩ 𝐴)) = ∅
8	3, 7	syl6eq 2701	. 2 ⊢ (𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
9		relxp 5160	. . . 4 ⊢ Rel (V × V)
10		restfn 16132	. . . . . 6 ⊢ ↾t Fn (V × V)
11		fndm 6028	. . . . . 6 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V))
12	10, 11	ax-mp 5	. . . . 5 ⊢ dom ↾t = (V × V)
13	12	releqi 5236	. . . 4 ⊢ (Rel dom ↾t ↔ Rel (V × V))
14	9, 13	mpbir 221	. . 3 ⊢ Rel dom ↾t
15	14	ovprc2 6725	. 2 ⊢ (¬ 𝐴 ∈ V → (∅ ↾t 𝐴) = ∅)
16	8, 15	pm2.61i 176	1 ⊢ (∅ ↾t 𝐴) = ∅

Syntax hints:   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606  ∅c0 3948   ↦ cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148   Fn wfn 5921  (class class class)co 6690   ↾_t crest 16128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-rest 16130

This theorem is referenced by:  firest  16140  topnval  16142  resstopn  21038  ussval  22110  bj-rest00  33159