Theorem crefss 29244

Description: The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Assertion

Ref	Expression
crefss	⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Proof of Theorem crefss

Dummy variables 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sslin 3801	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝒫 𝑗 ∩ 𝐴) ⊆ (𝒫 𝑗 ∩ 𝐵))
2		ssrexv 3630	. . . . . . 7 ⊢ ((𝒫 𝑗 ∩ 𝐴) ⊆ (𝒫 𝑗 ∩ 𝐵) → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))
4	3	imim2d 55	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) → (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)))
5	4	ralimdv 2946	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) → ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)))
6	5	anim2d 587	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)) → (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))))
7		eqid 2610	. . . 4 ⊢ ∪ 𝑗 = ∪ 𝑗
8	7	iscref 29239	. . 3 ⊢ (𝑗 ∈ CovHasRef𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)))
9	7	iscref 29239	. . 3 ⊢ (𝑗 ∈ CovHasRef𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)))
10	6, 8, 9	3imtr4g 284	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑗 ∈ CovHasRef𝐴 → 𝑗 ∈ CovHasRef𝐵))
11	10	ssrdv 3574	1 ⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  Topctop 20517  Refcref 21115  CovHasRefccref 29237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-cref 29238

This theorem is referenced by: (None)