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Mirrors > Home > ILE Home > Th. List > restrcl | GIF version |
Description: Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.) |
Ref | Expression |
---|---|
restrcl | ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0opn 13442 | . 2 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → ∅ ∈ (𝐽 ↾t 𝐴)) | |
2 | df-rest 12689 | . . 3 ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) | |
3 | 2 | elmpocl 6068 | . 2 ⊢ (∅ ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
4 | 1, 3 | syl 14 | 1 ⊢ ((𝐽 ↾t 𝐴) ∈ Top → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 ∅c0 3422 ↦ cmpt 4064 ran crn 4627 (class class class)co 5874 ↾t crest 12687 Topctop 13433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-rest 12689 df-top 13434 |
This theorem is referenced by: cnrest2r 13673 |
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