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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2lem | Structured version Visualization version GIF version |
Description: Lemma for equivbnd2 35085 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 5878 | . . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀 | |
3 | 1, 2 | eqsstri 4001 | . . . . 5 ⊢ 𝐷 ⊆ 𝑀 |
4 | dmss 5771 | . . . . 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 35074 | . . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌)) | |
7 | metf 22940 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ) | |
8 | fdm 6522 | . . . . . 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 484 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌)) |
11 | metf 22940 | . . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ) | |
12 | 11 | fdmd 6523 | . . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋)) |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋)) |
14 | 5, 10, 13 | 3sstr3d 4013 | . . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋)) |
15 | dmss 5771 | . . 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 5800 | . 2 ⊢ dom (𝑌 × 𝑌) = 𝑌 | |
18 | dmxpid 5800 | . 2 ⊢ dom (𝑋 × 𝑋) = 𝑋 | |
19 | 16, 17, 18 | 3sstr3g 4011 | 1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 dom cdm 5555 ↾ cres 5557 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 Metcmet 20531 Bndcbnd 35060 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
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-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 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-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-met 20539 df-bnd 35072 |
This theorem is referenced by: equivbnd2 35085 prdsbnd2 35088 cntotbnd 35089 cnpwstotbnd 35090 |
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