Theorem bj-restsn 37394

Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 37397 and bj-restsnid 37399. (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 5381	. . . 4 ⊢ {𝑌} ∈ V
2		elrest 17390	. . . 4 ⊢ (({𝑌} ∈ V ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴)))
3	1, 2	mpan 691	. . 3 ⊢ (𝐴 ∈ 𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴)))
4		velsn 4583	. . . . 5 ⊢ (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 = (𝑦 ∩ 𝐴))
5		ineq1 4153	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐴) = (𝑌 ∩ 𝐴))
6	5	sneqd 4579	. . . . . 6 ⊢ (𝑦 = 𝑌 → {(𝑦 ∩ 𝐴)} = {(𝑌 ∩ 𝐴)})
7	6	eleq2d 2822	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦 ∩ 𝐴)} ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
8	4, 7	bitr3id 285	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
9	8	rexsng 4620	. . 3 ⊢ (𝑌 ∈ 𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦 ∩ 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
10	3, 9	sylan9bbr 510	. 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌 ∩ 𝐴)}))
11	10	eqrdv 2734	1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429   ∩ cin 3888  {csn 4567  (class class class)co 7367   ↾_t crest 17383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-rest 17385

This theorem is referenced by:  bj-restsnss  37395  bj-restsnss2  37396