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Theorem bnd2lem 34198
Description: Lemma for equivbnd2 34199 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 5671 . . . . . 6 (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
31, 2eqsstri 3853 . . . . 5 𝐷𝑀
4 dmss 5568 . . . . 5 (𝐷𝑀 → dom 𝐷 ⊆ dom 𝑀)
53, 4mp1i 13 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 ⊆ dom 𝑀)
6 bndmet 34188 . . . . . 6 (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7 metf 22543 . . . . . 6 (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8 fdm 6299 . . . . . 6 (𝐷:(𝑌 × 𝑌)⟶ℝ → dom 𝐷 = (𝑌 × 𝑌))
96, 7, 83syl 18 . . . . 5 (𝐷 ∈ (Bnd‘𝑌) → dom 𝐷 = (𝑌 × 𝑌))
109adantl 475 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌))
11 metf 22543 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
1211fdmd 6300 . . . . 5 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
1312adantr 474 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
145, 10, 133sstr3d 3865 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15 dmss 5568 . . 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 5590 . 2 dom (𝑌 × 𝑌) = 𝑌
18 dmxpid 5590 . 2 dom (𝑋 × 𝑋) = 𝑋
1916, 17, 183sstr3g 3863 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2106  wss 3791   × cxp 5353  dom cdm 5355  cres 5357  wf 6131  cfv 6135  cr 10271  Metcmet 20128  Bndcbnd 34174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-met 20136  df-bnd 34186
This theorem is referenced by:  equivbnd2  34199  prdsbnd2  34202  cntotbnd  34203  cnpwstotbnd  34204
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