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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 34945 and bj-restsnss 34946. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsnid | ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3913 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | bj-restsnss 34946 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴}) | |
3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
4 | df-rest 16899 | . . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦))) | |
5 | 4 | reldmmpo 7333 | . . . 4 ⊢ Rel dom ↾t |
6 | 5 | ovprc2 7242 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅) |
7 | snprc 4623 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | 7 | biimpi 219 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
9 | 6, 8 | eqtr4d 2777 | . 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
10 | 3, 9 | pm2.61i 185 | 1 ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∩ cin 3856 ⊆ wss 3857 ∅c0 4227 {csn 4531 ↦ cmpt 5124 ran crn 5541 (class class class)co 7202 ↾t crest 16897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-rest 16899 |
This theorem is referenced by: (None) |
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