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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2lem | Structured version Visualization version GIF version |
Description: Lemma for equivbnd2 34132 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 5662 | . . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀 | |
3 | 1, 2 | eqsstri 3860 | . . . . 5 ⊢ 𝐷 ⊆ 𝑀 |
4 | dmss 5559 | . . . . 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 34121 | . . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌)) | |
7 | metf 22512 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ) | |
8 | fdm 6290 | . . . . . 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 22512 | . . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ) | |
12 | 11 | fdmd 6291 | . . . . 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 3872 | . . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | dmss 5559 | . . 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 5581 | . 2 ⊢ dom (𝑌 × 𝑌) = 𝑌 | |
18 | dmxpid 5581 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
19 | 16, 17, 18 | 3sstr3g 3870 | 1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 × cxp 5344 dom cdm 5346 ↾ cres 5348 ⟶wf 6123 ‘cfv 6127 ℝcr 10258 Metcmet 20099 Bndcbnd 34107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-met 20107 df-bnd 34119 |
This theorem is referenced by: equivbnd2 34132 prdsbnd2 34135 cntotbnd 34136 cnpwstotbnd 34137 |
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