Theorem bnd2lem 37757

Description: Lemma for equivbnd2 37758 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 5999	. . . . . 6 ⊢ (𝑀 ↾ (𝑌 × 𝑌)) ⊆ 𝑀
3	1, 2	eqsstri 4010	. . . . 5 ⊢ 𝐷 ⊆ 𝑀
4		dmss 5893	. . . . 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 37747	. . . . . 6 ⊢ (𝐷 ∈ (Bnd‘𝑌) → 𝐷 ∈ (Met‘𝑌))
7		metf 24285	. . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑌) → 𝐷:(𝑌 × 𝑌)⟶ℝ)
8		fdm 6725	. . . . . 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 24285	. . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ)
12	11	fdmd 6726	. . . . 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 4018	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑋))
15		dmss 5893	. . 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 5921	. 2 ⊢ dom (𝑌 × 𝑌) = 𝑌
18		dmxpid 5921	. 2 ⊢ dom (𝑋 × 𝑋) = 𝑋
19	16, 17, 18	3sstr3g 4016	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931   × cxp 5663  dom cdm 5665   ↾ cres 5667  ⟶wf 6537  ‘cfv 6541  ℝcr 11136  Metcmet 21312  Bndcbnd 37733

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8850  df-met 21320  df-bnd 37745

This theorem is referenced by:  equivbnd2  37758  prdsbnd2  37761  cntotbnd  37762  cnpwstotbnd  37763