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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2lem | Structured version Visualization version GIF version |
Description: Lemma for equivbnd2 35636 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 5861 | . . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀 | |
3 | 1, 2 | eqsstri 3921 | . . . . 5 ⊢ 𝐷 ⊆ 𝑀 |
4 | dmss 5756 | . . . . 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 35625 | . . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌)) | |
7 | metf 23182 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ) | |
8 | fdm 6532 | . . . . . 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 485 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌)) |
11 | metf 23182 | . . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ) | |
12 | 11 | fdmd 6534 | . . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋)) |
13 | 12 | adantr 484 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋)) |
14 | 5, 10, 13 | 3sstr3d 3933 | . . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | dmss 5756 | . . 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 5784 | . 2 ⊢ dom (𝑌 × 𝑌) = 𝑌 | |
18 | dmxpid 5784 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
19 | 16, 17, 18 | 3sstr3g 3931 | 1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 × cxp 5534 dom cdm 5536 ↾ cres 5538 ⟶wf 6354 ‘cfv 6358 ℝcr 10693 Metcmet 20303 Bndcbnd 35611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-met 20311 df-bnd 35623 |
This theorem is referenced by: equivbnd2 35636 prdsbnd2 35639 cntotbnd 35640 cnpwstotbnd 35641 |
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