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Theorem bnd2lem 35229
Description: Lemma for equivbnd2 35230 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)
Hypothesis
Ref Expression
bnd2lem.1 𝐷 = (𝑀 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
bnd2lem ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)

Proof of Theorem bnd2lem
StepHypRef Expression
1 bnd2lem.1 . . . . . 6 𝐷 = (𝑀 ↾ (𝑌 × 𝑌))
2 resss 5843 . . . . . 6 (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
31, 2eqsstri 3949 . . . . 5 𝐷𝑀
4 dmss 5735 . . . . 5 (𝐷𝑀 → dom 𝐷 ⊆ dom 𝑀)
53, 4mp1i 13 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 ⊆ dom 𝑀)
6 bndmet 35219 . . . . . 6 (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7 metf 22937 . . . . . 6 (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8 fdm 6495 . . . . . 6 (𝐷:(𝑌 × 𝑌)⟶ℝ → dom 𝐷 = (𝑌 × 𝑌))
96, 7, 83syl 18 . . . . 5 (𝐷 ∈ (Bnd‘𝑌) → dom 𝐷 = (𝑌 × 𝑌))
109adantl 485 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌))
11 metf 22937 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
1211fdmd 6497 . . . . 5 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
1312adantr 484 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
145, 10, 133sstr3d 3961 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15 dmss 5735 . . 3 ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
1614, 15syl 17 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
17 dmxpid 5764 . 2 dom (𝑌 × 𝑌) = 𝑌
18 dmxpid 5764 . 2 dom (𝑋 × 𝑋) = 𝑋
1916, 17, 183sstr3g 3959 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  wss 3881   × cxp 5517  dom cdm 5519  cres 5521  wf 6320  cfv 6324  cr 10525  Metcmet 20077  Bndcbnd 35205
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-met 20085  df-bnd 35217
This theorem is referenced by:  equivbnd2  35230  prdsbnd2  35233  cntotbnd  35234  cnpwstotbnd  35235
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