Theorem bj-restsnid 35258

Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 35253 and bj-restsnss 35254. (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 3943	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		bj-restsnss 35254	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
3	1, 2	mpan2 688	. 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4		df-rest 17133	. . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦)))
5	4	reldmmpo 7408	. . . 4 ⊢ Rel dom ↾t
6	5	ovprc2 7315	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7		snprc 4653	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8	7	biimpi 215	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
9	6, 8	eqtr4d 2781	. 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
10	3, 9	pm2.61i 182	1 ⊢ ({𝐴} ↾t 𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  {csn 4561   ↦ cmpt 5157  ran crn 5590  (class class class)co 7275   ↾_t crest 17131

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-rest 17133

This theorem is referenced by: (None)