Theorem bnd2lem 37792

Description: Lemma for equivbnd2 37793 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 6026	. . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
3	1, 2	eqsstri 4033	. . . . 5 ⊢ 𝐷 ⊆ 𝑀
4		dmss 5920	. . . . 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 37782	. . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7		metf 24365	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8		fdm 6753	. . . . . 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 481	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝐷 = (𝑌 × 𝑌))
11		metf 24365	. . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
12	11	fdmd 6754	. . . . 5 ⊢ (𝑀 ∈ (Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
13	12	adantr 480	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → dom 𝑀 = (𝑋 × 𝑋))
14	5, 10, 13	3sstr3d 4045	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15		dmss 5920	. . 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 5948	. 2 ⊢ dom (𝑌 × 𝑌) = 𝑌
18		dmxpid 5948	. 2 ⊢ dom (𝑋 × 𝑋) = 𝑋
19	16, 17, 18	3sstr3g 4043	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3966   × cxp 5691  dom cdm 5693   ↾ cres 5695  ⟶wf 6565  ‘cfv 6569  ℝcr 11161  Metcmet 21377  Bndcbnd 37768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761  ax-cnex 11218  ax-resscn 11219

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577  df-ov 7441  df-oprab 7442  df-mpo 7443  df-map 8876  df-met 21385  df-bnd 37780

This theorem is referenced by:  equivbnd2  37793  prdsbnd2  37796  cntotbnd  37797  cnpwstotbnd  37798