Theorem bj-rest0 33171

Description: An elementwise intersection on a family containing the empty set contains the empty set. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-rest0	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))

Proof of Theorem bj-rest0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		in0 4001	. . . . 5 ⊢ (𝐴 ∩ ∅) = ∅
2		incom 3838	. . . . 5 ⊢ (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
3	1, 2	eqtr3i 2675	. . . 4 ⊢ ∅ = (∅ ∩ 𝐴)
4		0ex 4823	. . . . 5 ⊢ ∅ ∈ V
5		eleq1 2718	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑋 ↔ ∅ ∈ 𝑋))
6		ineq1 3840	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝐴) = (∅ ∩ 𝐴))
7	6	eqeq2d 2661	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ = (𝑥 ∩ 𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
8	5, 7	anbi12d 747	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
9	4, 8	spcev 3331	. . . 4 ⊢ ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
10	3, 9	mpan2 707	. . 3 ⊢ (∅ ∈ 𝑋 → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
11		df-rex 2947	. . 3 ⊢ (∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
12	10, 11	sylibr 224	. 2 ⊢ (∅ ∈ 𝑋 → ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴))
13		elrest 16135	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴)))
14	12, 13	syl5ibr 236	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942   ∩ cin 3606  ∅c0 3948  (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-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-rest 16130

This theorem is referenced by: (None)