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Theorem bnd2lem 33219
 Description: Lemma for equivbnd2 33220 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 5381 . . . . . 6 (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
31, 2eqsstri 3614 . . . . 5 𝐷𝑀
4 dmss 5283 . . . . 5 (𝐷𝑀 → dom 𝐷 ⊆ dom 𝑀)
53, 4mp1i 13 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 ⊆ dom 𝑀)
6 bndmet 33209 . . . . . 6 (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7 metf 22045 . . . . . 6 (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8 fdm 6008 . . . . . 6 (𝐷:(𝑌 × 𝑌)⟶ℝ → dom 𝐷 = (𝑌 × 𝑌))
96, 7, 83syl 18 . . . . 5 (𝐷 ∈ (Bnd‘𝑌) → dom 𝐷 = (𝑌 × 𝑌))
109adantl 482 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌))
11 metf 22045 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
12 fdm 6008 . . . . . 6 (𝑀:(𝑋 × 𝑋)⟶ℝ → dom 𝑀 = (𝑋 × 𝑋))
1311, 12syl 17 . . . . 5 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
1413adantr 481 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
155, 10, 143sstr3d 3626 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
16 dmss 5283 . . 3 ((𝑌 × 𝑌) ⊆ (𝑋 × 𝑋) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
1715, 16syl 17 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom (𝑌 × 𝑌) ⊆ dom (𝑋 × 𝑋))
18 dmxpid 5305 . 2 dom (𝑌 × 𝑌) = 𝑌
19 dmxpid 5305 . 2 dom (𝑋 × 𝑋) = 𝑋
2017, 18, 193sstr3g 3624 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ⊆ wss 3555   × cxp 5072  dom cdm 5074   ↾ cres 5076  ⟶wf 5843  ‘cfv 5847  ℝcr 9879  Metcme 19651  Bndcbnd 33195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-met 19659  df-bnd 33207 This theorem is referenced by:  equivbnd2  33220  prdsbnd2  33223  cntotbnd  33224  cnpwstotbnd  33225
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