Theorem iscref 29911

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
iscref.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iscref	⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))

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

Allowed substitution hints:   𝑋(𝑦,𝑧)

Proof of Theorem iscref

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4161	. . 3 ⊢ (𝑗 = 𝐽 → 𝒫 𝑗 = 𝒫 𝐽)
2		unieq 4444	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
3		iscref.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4	2, 3	syl6eqr 2674	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
5	4	eqeq1d 2624	. . . 4 ⊢ (𝑗 = 𝐽 → (∪ 𝑗 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
6	1	ineq1d 3813	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝒫 𝑗 ∩ 𝐴) = (𝒫 𝐽 ∩ 𝐴))
7	6	rexeqdv 3145	. . . 4 ⊢ (𝑗 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 ↔ ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦))
8	5, 7	imbi12d 334	. . 3 ⊢ (𝑗 = 𝐽 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
9	1, 8	raleqbidv 3152	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))
10		df-cref 29910	. 2 ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)}
11	9, 10	elrab2 3366	1 ⊢ (𝐽 ∈ CovHasRef𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝑦)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653  Topctop 20698  Refcref 21305  CovHasRefccref 29909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-cref 29910

This theorem is referenced by:  creftop  29913  crefi  29914  crefss  29916  cmpcref  29917  cmppcmp  29925  dispcmp  29926  pcmplfin  29927