Theorem bj-restsnid 37110

Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 37105 and bj-restsnss 37106. (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 3986	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		bj-restsnss 37106	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
3	1, 2	mpan2 691	. 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4		df-rest 17441	. . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦)))
5	4	reldmmpo 7546	. . . 4 ⊢ Rel dom ↾t
6	5	ovprc2 7450	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7		snprc 4698	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8	7	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
9	6, 8	eqtr4d 2774	. 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 2109  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606   ↦ cmpt 5206  ran crn 5660  (class class class)co 7410   ↾_t crest 17439

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  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 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-rest 17441

This theorem is referenced by: (None)