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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-restsnss | Structured version Visualization version GIF version |
Description: Special case of bj-restsn 34376. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
bj-restsnss | ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqin2 4192 | . . 3 ⊢ (𝐴 ⊆ 𝑌 ↔ (𝑌 ∩ 𝐴) = 𝐴) | |
2 | sneq 4577 | . . 3 ⊢ ((𝑌 ∩ 𝐴) = 𝐴 → {(𝑌 ∩ 𝐴)} = {𝐴}) | |
3 | 1, 2 | sylbi 219 | . 2 ⊢ (𝐴 ⊆ 𝑌 → {(𝑌 ∩ 𝐴)} = {𝐴}) |
4 | ssexg 5227 | . . . 4 ⊢ ((𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐴 ∈ V) | |
5 | 4 | ancoms 461 | . . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ V) |
6 | bj-restsn 34376 | . . 3 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ∈ V) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) | |
7 | 5, 6 | syldan 593 | . 2 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) |
8 | eqeq2 2833 | . . 3 ⊢ ({(𝑌 ∩ 𝐴)} = {𝐴} → (({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)} ↔ ({𝑌} ↾t 𝐴) = {𝐴})) | |
9 | 8 | biimpa 479 | . 2 ⊢ (({(𝑌 ∩ 𝐴)} = {𝐴} ∧ ({𝑌} ↾t 𝐴) = {(𝑌 ∩ 𝐴)}) → ({𝑌} ↾t 𝐴) = {𝐴}) |
10 | 3, 7, 9 | syl2an2 684 | 1 ⊢ ((𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌) → ({𝑌} ↾t 𝐴) = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 {csn 4567 (class class class)co 7156 ↾t crest 16694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-rest 16696 |
This theorem is referenced by: bj-restsn10 34380 bj-restsnid 34381 |
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