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Theorem bnd2lem 33720
Description: Lemma for equivbnd2 33721 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 5457 . . . . . 6 (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
31, 2eqsstri 3668 . . . . 5 𝐷𝑀
4 dmss 5355 . . . . 5 (𝐷𝑀 → dom 𝐷 ⊆ dom 𝑀)
53, 4mp1i 13 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 ⊆ dom 𝑀)
6 bndmet 33710 . . . . . 6 (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7 metf 22182 . . . . . 6 (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8 fdm 6089 . . . . . 6 (𝐷:(𝑌 × 𝑌)⟶ℝ → dom 𝐷 = (𝑌 × 𝑌))
96, 7, 83syl 18 . . . . 5 (𝐷 ∈ (Bnd‘𝑌) → dom 𝐷 = (𝑌 × 𝑌))
109adantl 481 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌))
11 metf 22182 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
12 fdm 6089 . . . . . 6 (𝑀:(𝑋 × 𝑋)⟶ℝ → dom 𝑀 = (𝑋 × 𝑋))
1311, 12syl 17 . . . . 5 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
1413adantr 480 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
155, 10, 143sstr3d 3680 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
16 dmss 5355 . . 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 5377 . 2 dom (𝑌 × 𝑌) = 𝑌
19 dmxpid 5377 . 2 dom (𝑋 × 𝑋) = 𝑋
2017, 18, 193sstr3g 3678 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wss 3607   × cxp 5141  dom cdm 5143  cres 5145  wf 5922  cfv 5926  cr 9973  Metcme 19780  Bndcbnd 33696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-met 19788  df-bnd 33708
This theorem is referenced by:  equivbnd2  33721  prdsbnd2  33724  cntotbnd  33725  cnpwstotbnd  33726
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