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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2lem | Structured version Visualization version GIF version |
Description: Lemma for equivbnd2 36240 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 5961 | . . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀 | |
3 | 1, 2 | eqsstri 3977 | . . . . 5 ⊢ 𝐷 ⊆ 𝑀 |
4 | dmss 5857 | . . . . 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 36229 | . . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌)) | |
7 | metf 23679 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ) | |
8 | fdm 6675 | . . . . . 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 482 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌)) |
11 | metf 23679 | . . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ) | |
12 | 11 | fdmd 6677 | . . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋)) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋)) |
14 | 5, 10, 13 | 3sstr3d 3989 | . . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | dmss 5857 | . . 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 5884 | . 2 ⊢ dom (𝑌 × 𝑌) = 𝑌 | |
18 | dmxpid 5884 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
19 | 16, 17, 18 | 3sstr3g 3987 | 1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3909 × cxp 5630 dom cdm 5632 ↾ cres 5634 ⟶wf 6490 ‘cfv 6494 ℝcr 11047 Metcmet 20778 Bndcbnd 36215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-map 8764 df-met 20786 df-bnd 36227 |
This theorem is referenced by: equivbnd2 36240 prdsbnd2 36243 cntotbnd 36244 cnpwstotbnd 36245 |
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