Theorem bj-restsn 37817

Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 37820 and bj-restsnid 37822. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restsn	⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})

Proof of Theorem bj-restsn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5408	. . . 4 ⊢ {𝑌} ∈ V
2		elrest 17514	. . . 4 ⊢ (({𝑌} ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	mpan 703	. . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴)))
4		velsn 4603	. . . . 5 ⊢ (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 = (𝑦 ∩ 𝐴))
5		ineq1 4162	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐴) = (𝑌 ∩ 𝐴))
6	5	sneqd 4599	. . . . . 6 ⊢ (𝑦 = 𝑌 → {(𝑦 ∩ 𝐴)} = {(𝑌 ∩ 𝐴)})
7	6	eleq2d 2848	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
8	4, 7	bitr3id 288	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
9	8	rexsng 4640	. . 3 ⊢ (𝑌 ∈ 𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
10	3, 9	sylan9bbr 520	. 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
11	10	eqrdv 2760	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453   ∩ cin 3901  {csn 4587  (class class class)co 7416   ↾_t crest 17507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-rest 17509

This theorem is used by:  bj-restsnss  37818  bj-restsnss2  37819