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Mirrors > Home > MPE Home > Th. List > ssrest | Structured version Visualization version GIF version |
Description: If 𝐾 is a finer topology than 𝐽, then the subspace topologies induced by 𝐴 maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
ssrest | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝐴) ⊆ (𝐾 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝑥 ∈ (𝐽 ↾t 𝐴)) | |
2 | ssrexv 4034 | . . . . . 6 ⊢ (𝐽 ⊆ 𝐾 → (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) | |
3 | 2 | ad2antlr 725 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴) → ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) |
4 | n0i 4299 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → ¬ (𝐽 ↾t 𝐴) = ∅) | |
5 | restfn 16698 | . . . . . . . . . 10 ⊢ ↾t Fn (V × V) | |
6 | fndm 6455 | . . . . . . . . . 10 ⊢ ( ↾t Fn (V × V) → dom ↾t = (V × V)) | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . 9 ⊢ dom ↾t = (V × V) |
8 | 7 | ndmov 7332 | . . . . . . . 8 ⊢ (¬ (𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ∅) |
9 | 4, 8 | nsyl2 143 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐽 ↾t 𝐴) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
10 | 9 | adantl 484 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝐽 ∈ V ∧ 𝐴 ∈ V)) |
11 | elrest 16701 | . . . . . 6 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝑥 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐽 𝑥 = (𝑦 ∩ 𝐴))) |
13 | simpll 765 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝐾 ∈ 𝑉) | |
14 | 10 | simprd 498 | . . . . . 6 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝐴 ∈ V) |
15 | elrest 16701 | . . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐾 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) | |
16 | 13, 14, 15 | syl2anc 586 | . . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝑥 ∈ (𝐾 ↾t 𝐴) ↔ ∃𝑦 ∈ 𝐾 𝑥 = (𝑦 ∩ 𝐴))) |
17 | 3, 12, 16 | 3imtr4d 296 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ (𝐾 ↾t 𝐴))) |
18 | 1, 17 | mpd 15 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐽 ↾t 𝐴)) → 𝑥 ∈ (𝐾 ↾t 𝐴)) |
19 | 18 | ex 415 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐽 ↾t 𝐴) → 𝑥 ∈ (𝐾 ↾t 𝐴))) |
20 | 19 | ssrdv 3973 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐾) → (𝐽 ↾t 𝐴) ⊆ (𝐾 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 × cxp 5553 dom cdm 5555 Fn wfn 6350 (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-pow 5266 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-1st 7689 df-2nd 7690 df-rest 16696 |
This theorem is referenced by: 1stcrest 22061 kgencmp 22153 kgencmp2 22154 kgen2ss 22163 ssufl 22526 cnfsmf 43037 smfsssmf 43040 |
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