Theorem crefi 30512

Description: The property that every open cover has an 𝐴 refinement for the topological space 𝐽. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Hypothesis

Ref	Expression
crefi.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
crefi	⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐽   𝑧,𝐶

Allowed substitution hint:   𝑋(𝑧)

Proof of Theorem crefi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1127	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐽 ∈ CovHasRef𝐴)
2		simp2 1128	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ⊆ 𝐽)
3	1, 2	sselpwd 5044	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝐶 ∈ 𝒫 𝐽)
4		crefi.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
5	4	iscref 30509	. . . 4 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
6	5	simprbi 492	. . 3 ⊢ (𝐽 ∈ CovHasRef𝐴 → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
7	6	3ad2ant1 1124	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8		simp3 1129	. 2 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → 𝑋 = ∪ 𝐶)
9		unieq 4679	. . . . 5 ⊢ (𝑦 = 𝐶 → ∪ 𝑦 = ∪ 𝐶)
10	9	eqeq2d 2788	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐶))
11		breq2 4890	. . . . 5 ⊢ (𝑦 = 𝐶 → (𝑧Ref𝑦 ↔ 𝑧Ref𝐶))
12	11	rexbidv 3237	. . . 4 ⊢ (𝑦 = 𝐶 → (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶))
13	10, 12	imbi12d 336	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
14	13	rspcv 3507	. 2 ⊢ (𝐶 ∈ 𝒫 𝐽 → (∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦) → (𝑋 = ∪ 𝐶 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)))
15	3, 7, 8, 14	syl3c 66	1 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)

Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091   ∩ cin 3791   ⊆ wss 3792  𝒫 cpw 4379  ∪ cuni 4671   class class class wbr 4886  Topctop 21105  Refcref 21714  CovHasRefccref 30507

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 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017

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-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-cref 30508

This theorem is referenced by:  crefdf  30513