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)) → 𝑌 ⊆ 𝑋)

Proof of Theorem bnd2lem

Step	Hyp	Ref	Expression
1		bnd2lem.1	. . . . . 6 ⊢ 𝐷 = (𝑀 ↾ (𝑌 × 𝑌))
2		resss 5671	. . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
3	1, 2	eqsstri 3853	. . . . 5 ⊢ 𝐷 ⊆ 𝑀
4		dmss 5568	. . . . 5 ⊢ (𝐷 ⊆ 𝑀 → dom 𝐷 ⊆ dom 𝑀)
5	3, 4	mp1i 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))
9	6, 7, 8	3syl 18	. . . . 5 ⊢ (𝐷 ∈ (Bnd‘𝑌) → dom 𝐷 = (𝑌 × 𝑌))
10	9	adantl 475	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌))
11		metf 22543	. . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
12	11	fdmd 6300	. . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
13	12	adantr 474	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
14	5, 10, 13	3sstr3d 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))
16	14, 15	syl 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
19	16, 17, 18	3sstr3g 3863	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106   ⊆ wss 3791   × cxp 5353  dom cdm 5355   ↾ cres 5357  ⟶wf 6131  ‘cfv 6135  ℝcr 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