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Theorem bnd2lem 34131
Description: Lemma for equivbnd2 34132 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 5662 . . . . . 6 (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
31, 2eqsstri 3860 . . . . 5 𝐷𝑀
4 dmss 5559 . . . . 5 (𝐷𝑀 → dom 𝐷 ⊆ dom 𝑀)
53, 4mp1i 13 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 ⊆ dom 𝑀)
6 bndmet 34121 . . . . . 6 (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7 metf 22512 . . . . . 6 (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8 fdm 6290 . . . . . 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 22512 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
1211fdmd 6291 . . . . 5 (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
1312adantr 474 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
145, 10, 133sstr3d 3872 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15 dmss 5559 . . 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 5581 . 2 dom (𝑌 × 𝑌) = 𝑌
18 dmxpid 5581 . 2 dom (𝑋 × 𝑋) = 𝑋
1916, 17, 183sstr3g 3870 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wss 3798   × cxp 5344  dom cdm 5346  cres 5348  wf 6123  cfv 6127  cr 10258  Metcmet 20099  Bndcbnd 34107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-map 8129  df-met 20107  df-bnd 34119
This theorem is referenced by:  equivbnd2  34132  prdsbnd2  34135  cntotbnd  34136  cnpwstotbnd  34137
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