Theorem crefdf 33808

Description: A formulation of crefi 33807 easier to use for definitions. (Contributed by Thierry Arnoux, 7-Jan-2020.)

Hypotheses

Ref	Expression
crefi.x	⊢ 𝑋 = ∪ 𝐽
crefdf.b	⊢ 𝐵 = CovHasRef𝐴
crefdf.p	⊢ (𝑧 ∈ 𝐴 → 𝜑)

Assertion

Ref	Expression
crefdf	⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))

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

Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑧)   𝑋(𝑧)

Proof of Theorem crefdf

Step	Hyp	Ref	Expression
1		crefdf.b	. . . 4 ⊢ 𝐵 = CovHasRef𝐴
2	1	eleq2i 2830	. . 3 ⊢ (𝐽 ∈ 𝐵 ↔ 𝐽 ∈ CovHasRef𝐴)
3		crefi.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
4	3	crefi 33807	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
5	2, 4	syl3an1b 1402	. 2 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
6		elin 3978	. . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴))
7		crefdf.p	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝜑)
8	7	anim2i 617	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑))
9	6, 8	sylbi 217	. . . . 5 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑))
10	9	anim1i 615	. . . 4 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶))
11		anass 468	. . . 4 ⊢ (((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶)))
12	10, 11	sylib 218	. . 3 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶)))
13	12	reximi2 3076	. 2 ⊢ (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶 → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))
14	5, 13	syl 17	1 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∃wrex 3067   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  ∪ cuni 4911   class class class wbr 5147  Refcref 23525  CovHasRefccref 33802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-cref 33803

This theorem is referenced by:  cmpfiref  33811  ldlfcntref  33814