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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-restsnid | Structured version Visualization version GIF version |
Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 35901 and bj-restsnss 35902. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsnid | ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4003 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | bj-restsnss 35902 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴}) | |
3 | 1, 2 | mpan2 690 | . 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
4 | df-rest 17364 | . . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦))) | |
5 | 4 | reldmmpo 7538 | . . . 4 ⊢ Rel dom ↾t |
6 | 5 | ovprc2 7444 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅) |
7 | snprc 4720 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | 7 | biimpi 215 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
9 | 6, 8 | eqtr4d 2776 | . 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
10 | 3, 9 | pm2.61i 182 | 1 ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 {csn 4627 ↦ cmpt 5230 ran crn 5676 (class class class)co 7404 ↾t crest 17362 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-rest 17364 |
This theorem is referenced by: (None) |
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