Theorem bj-restsnid 37088

Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 37083 and bj-restsnss 37084. (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
bj-restsnid	⊢ ({𝐴} ↾t 𝐴) = {𝐴}

Proof of Theorem bj-restsnid

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

Step	Hyp	Ref	Expression
1		ssid 4006	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		bj-restsnss 37084	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
3	1, 2	mpan2 691	. 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4		df-rest 17467	. . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦)))
5	4	reldmmpo 7567	. . . 4 ⊢ Rel dom ↾t
6	5	ovprc2 7471	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7		snprc 4717	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8	7	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
9	6, 8	eqtr4d 2780	. 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
10	3, 9	pm2.61i 182	1 ⊢ ({𝐴} ↾t 𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  {csn 4626   ↦ cmpt 5225  ran crn 5686  (class class class)co 7431   ↾_t crest 17465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467

This theorem is referenced by: (None)