Theorem crefdf 30756

Description: A formulation of crefi 30755 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 2851	. . 3 ⊢ (𝐽 ∈ 𝐵 ↔ 𝐽 ∈ CovHasRef𝐴)
3		crefi.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
4	3	crefi 30755	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
5	2, 4	syl3an1b 1383	. 2 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
6		elin 4051	. . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴))
7		crefdf.p	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝜑)
8	7	anim2i 607	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑))
9	6, 8	sylbi 209	. . . . 5 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑))
10	9	anim1i 605	. . . 4 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶))
11		anass 461	. . . 4 ⊢ (((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶)))
12	10, 11	sylib 210	. . 3 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶)))
13	12	reximi2 3185	. 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 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ∃wrex 3083   ∩ cin 3822   ⊆ wss 3823  𝒫 cpw 4416  ∪ cuni 4706   class class class wbr 4923  Refcref 21808  CovHasRefccref 30750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744  ax-sep 5054

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-cref 30751

This theorem is referenced by:  cmpfiref  30759  ldlfcntref  30762