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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2lem | Structured version Visualization version GIF version |
Description: Lemma for equivbnd2 34199 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.) |
Ref | Expression |
---|---|
bnd2lem.1 | ⊢ 𝐷 = (𝑀 ↾ (𝑌 × 𝑌)) |
Ref | Expression |
---|---|
bnd2lem | ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnd2lem.1 | . . . . . 6 ⊢ 𝐷 = (𝑀 ↾ (𝑌 × 𝑌)) | |
2 | resss 5671 | . . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀 | |
3 | 1, 2 | eqsstri 3853 | . . . . 5 ⊢ 𝐷 ⊆ 𝑀 |
4 | dmss 5568 | . . . . 5 ⊢ (𝐷 ⊆ 𝑀 → dom 𝐷 ⊆ dom 𝑀) | |
5 | 3, 4 | mp1i 13 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 ⊆ dom 𝑀) |
6 | bndmet 34188 | . . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌)) | |
7 | metf 22543 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ) | |
8 | fdm 6299 | . . . . . 6 ⊢ (𝐷:(𝑌 × 𝑌)⟶ℝ → dom 𝐷 = (𝑌 × 𝑌)) | |
9 | 6, 7, 8 | 3syl 18 | . . . . 5 ⊢ (𝐷 ∈ (Bnd‘𝑌) → dom 𝐷 = (𝑌 × 𝑌)) |
10 | 9 | adantl 475 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌)) |
11 | metf 22543 | . . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ) | |
12 | 11 | fdmd 6300 | . . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋)) |
13 | 12 | adantr 474 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋)) |
14 | 5, 10, 13 | 3sstr3d 3865 | . . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | dmss 5568 | . . 3 ⊢ ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋)) | |
16 | 14, 15 | syl 17 | . 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋)) |
17 | dmxpid 5590 | . 2 ⊢ dom (𝑌 × 𝑌) = 𝑌 | |
18 | dmxpid 5590 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
19 | 16, 17, 18 | 3sstr3g 3863 | 1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 × cxp 5353 dom cdm 5355 ↾ cres 5357 ⟶wf 6131 ‘cfv 6135 ℝcr 10271 Metcmet 20128 Bndcbnd 34174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-met 20136 df-bnd 34186 |
This theorem is referenced by: equivbnd2 34199 prdsbnd2 34202 cntotbnd 34203 cnpwstotbnd 34204 |
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