Theorem bj-rest0 35911

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 4389	. . . . 5 ⊢ (𝐴 ∩ ∅) = ∅
2		incom 4199	. . . . 5 ⊢ (𝐴 ∩ ∅) = (∅ ∩ 𝐴)
3	1, 2	eqtr3i 2763	. . . 4 ⊢ ∅ = (∅ ∩ 𝐴)
4		0ex 5305	. . . . 5 ⊢ ∅ ∈ V
5		eleq1 2822	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑋 ↔ ∅ ∈ 𝑋))
6		ineq1 4203	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑥 ∩ 𝐴) = (∅ ∩ 𝐴))
7	6	eqeq2d 2744	. . . . . 6 ⊢ (𝑥 = ∅ → (∅ = (𝑥 ∩ 𝐴) ↔ ∅ = (∅ ∩ 𝐴)))
8	5, 7	anbi12d 632	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)) ↔ (∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴))))
9	4, 8	spcev 3595	. . . 4 ⊢ ((∅ ∈ 𝑋 ∧ ∅ = (∅ ∩ 𝐴)) → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
10	3, 9	mpan2 690	. . 3 ⊢ (∅ ∈ 𝑋 → ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
11		df-rex 3072	. . 3 ⊢ (∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑋 ∧ ∅ = (𝑥 ∩ 𝐴)))
12	10, 11	sylibr 233	. 2 ⊢ (∅ ∈ 𝑋 → ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴))
13		elrest 17368	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ (𝑋 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝑋 ∅ = (𝑥 ∩ 𝐴)))
14	12, 13	imbitrrid 245	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (∅ ∈ 𝑋 → ∅ ∈ (𝑋 ↾t 𝐴)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃wrex 3071   ∩ cin 3945  ∅c0 4320  (class class class)co 7403   ↾_t crest 17361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pr 5425  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-rest 17363

This theorem is referenced by: (None)