![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 36453 and bj-restsnss 36454. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsnid | ⊢ ({𝐴} ↾t 𝐴) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3996 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
2 | bj-restsnss 36454 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝐴) → ({𝐴} ↾t 𝐴) = {𝐴}) | |
3 | 1, 2 | mpan2 688 | . 2 ⊢ (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴}) |
4 | df-rest 17367 | . . . . 5 ⊢ ↾t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧 ∈ 𝑥 ↦ (𝑧 ∩ 𝑦))) | |
5 | 4 | reldmmpo 7535 | . . . 4 ⊢ Rel dom ↾t |
6 | 5 | ovprc2 7441 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅) |
7 | snprc 4713 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | 7 | biimpi 215 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
9 | 6, 8 | eqtr4d 2767 | . 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 1533 ∈ wcel 2098 Vcvv 3466 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 {csn 4620 ↦ cmpt 5221 ran crn 5667 (class class class)co 7401 ↾t crest 17365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-rest 17367 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |