Theorem cover2g 32679

Description: Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑." Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)

Hypothesis

Ref	Expression
cover2g.1	⊢ 𝐴 = ∪ 𝐵

Assertion

Ref	Expression
cover2g	⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))

Distinct variable groups:   𝜑,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐴,𝑧

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem cover2g

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4380	. . . 4 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = ∪ 𝐵)
2		cover2g.1	. . . 4 ⊢ 𝐴 = ∪ 𝐵
3	1, 2	syl6eqr 2662	. . 3 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = 𝐴)
4		rexeq 3116	. . 3 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
5	3, 4	raleqbidv 3129	. 2 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ ∪ 𝑏∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
6		pweq 4111	. . 3 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
7	3	eqeq2d 2620	. . . 4 ⊢ (𝑏 = 𝐵 → (∪ 𝑧 = ∪ 𝑏 ↔ ∪ 𝑧 = 𝐴))
8	7	anbi1d 737	. . 3 ⊢ (𝑏 = 𝐵 → ((∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ (∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
9	6, 8	rexeqbidv 3130	. 2 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏(∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
10		vex 3176	. . 3 ⊢ 𝑏 ∈ V
11		eqid 2610	. . 3 ⊢ ∪ 𝑏 = ∪ 𝑏
12	10, 11	cover2 32678	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑏∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏(∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑))
13	5, 9, 12	vtoclbg 3240	1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  𝒫 cpw 4108  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373

This theorem is referenced by: (None)