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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-restsnss2 | Structured version Visualization version GIF version |
Description: Special case of bj-restsn 34367. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsnss2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {𝑌}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3952 | . . 3 ⊢ (𝑌 ⊆ 𝐴 ↔ (𝑌 ∩ 𝐴) = 𝑌) | |
2 | sneq 4571 | . . 3 ⊢ ((𝑌 ∩ 𝐴) = 𝑌 → {(𝑌 ∩ 𝐴)} = {𝑌}) | |
3 | 1, 2 | sylbi 219 | . 2 ⊢ (𝑌 ⊆ 𝐴 → {(𝑌 ∩ 𝐴)} = {𝑌}) |
4 | ssexg 5220 | . . . 4 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑌 ∈ V) | |
5 | 4 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → 𝑌 ∈ V) |
6 | bj-restsn 34367 | . . . 4 ⊢ ((𝑌 ∈ V ∧ 𝐴 ∈ 𝑉) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) | |
7 | 6 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ V) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
8 | 5, 7 | syldan 593 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
9 | eqeq2 2833 | . . 3 ⊢ ({(𝑌 ∩ 𝐴)} = {𝑌} → (({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)} ↔ ({𝑌} ↾t 𝐴) = {𝑌})) | |
10 | 9 | biimpa 479 | . 2 ⊢ (({(𝑌 ∩ 𝐴)} = {𝑌} ∧ ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) → ({𝑌} ↾t 𝐴) = {𝑌}) |
11 | 3, 8, 10 | syl2an2 684 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴) → ({𝑌} ↾t 𝐴) = {𝑌}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 {csn 4561 (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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-rest 16690 |
This theorem is referenced by: bj-restsn0 34370 |
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