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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 34367 and bj-restsnss 34368. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsnid | ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3988 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | bj-restsnss 34368 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴}) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
4 | df-rest 16690 | . . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦))) | |
5 | 4 | reldmmpo 7279 | . . . 4 ⊢ Rel dom ↾t |
6 | 5 | ovprc2 7190 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅) |
7 | snprc 4646 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | 7 | biimpi 218 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
9 | 6, 8 | eqtr4d 2859 | . 2 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
10 | 3, 9 | pm2.61i 184 | 1 ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 {csn 4560 ↦ cmpt 5138 ran crn 5550 (class class class)co 7150 ↾t crest 16688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-rest 16690 |
This theorem is referenced by: (None) |
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