Theorem crefdf 34269

Description: A formulation of crefi 34268 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 2858	. . 3 ⊢ (𝐽 ∈ 𝐵 ↔ 𝐽 ∈ CovHasRef𝐴)
3		crefi.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
4	3	crefi 34268	. . 3 ⊢ ((𝐽 ∈ CovHasRef𝐴 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
5	2, 4	syl3an1b 1430	. 2 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶)
6		elin 3924	. . . . . 6 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴))
7		crefdf.p	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝜑)
8	7	anim2i 629	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 𝐽 ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑))
9	6, 8	sylbi 220	. . . . 5 ⊢ (𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) → (𝑧 ∈ 𝒫 𝐽 ∧ 𝜑))
10	9	anim1i 627	. . . 4 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → ((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶))
11		anass 474	. . . 4 ⊢ (((𝑧 ∈ 𝒫 𝐽 ∧ 𝜑) ∧ 𝑧Ref𝐶) ↔ (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶)))
12	10, 11	sylib 221	. . 3 ⊢ ((𝑧 ∈ (𝒫 𝐽 ∩ 𝐴) ∧ 𝑧Ref𝐶) → (𝑧 ∈ 𝒫 𝐽 ∧ (𝜑 ∧ 𝑧Ref𝐶)))
13	12	reximi2 3101	. 2 ⊢ (∃𝑧 ∈ (𝒫 𝐽 ∩ 𝐴)𝑧Ref𝐶 → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))
14	5, 13	syl 18	1 ⊢ ((𝐽 ∈ 𝐵 ∧ 𝐶 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝐶) → ∃𝑧 ∈ 𝒫 𝐽(𝜑 ∧ 𝑧Ref𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3092   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877   class class class wbr 5114  Refcref 23696  CovHasRefccref 34263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-cref 34264

This theorem is used by:  cmpfiref  34272  ldlfcntref  34275