Theorem bj-restsnid 37574

Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 37569 and bj-restsnss 37570. (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 3958	. . 3 ⊢ 𝐴 ⊆ 𝐴
2		bj-restsnss 37570	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
3	1, 2	mpan2 701	. 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4		df-rest 17451	. . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦)))
5	4	reldmmpo 7530	. . . 4 ⊢ Rel dom ↾t
6	5	ovprc2 7436	. . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7		snprc 4676	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
8	7	biimpi 218	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
9	6, 8	eqtr4d 2800	. 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
10	3, 9	pm2.61i 183	1 ⊢ ({𝐴} ↾t 𝐴) = {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582   ↦ cmpt 5181  ran crn 5648  (class class class)co 7396   ↾_t crest 17449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-rest 17451

This theorem is referenced by: (None)