Theorem cover2g 37676

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 4942	. . . 4 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = ∪ 𝐵)
2		cover2g.1	. . . 4 ⊢ 𝐴 = ∪ 𝐵
3	1, 2	eqtr4di 2798	. . 3 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = 𝐴)
4		rexeq 3330	. . 3 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
5	3, 4	raleqbidv 3354	. 2 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ ∪ 𝑏∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑)))
6		pweq 4636	. . 3 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
7	3	eqeq2d 2751	. . . 4 ⊢ (𝑏 = 𝐵 → (∪ 𝑧 = ∪ 𝑏 ↔ ∪ 𝑧 = 𝐴))
8	7	anbi1d 630	. . 3 ⊢ (𝑏 = 𝐵 → ((∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ (∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
9	6, 8	rexeqbidv 3355	. 2 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏(∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
10		vex 3492	. . 3 ⊢ 𝑏 ∈ V
11		eqid 2740	. . 3 ⊢ ∪ 𝑏 = ∪ 𝑏
12	10, 11	cover2 37675	. 2 ⊢ (∀𝑥 ∈ ∪ 𝑏∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏(∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑))
13	5, 9, 12	vtoclbg 3569	1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  𝒫 cpw 4622  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624  df-uni 4932

This theorem is referenced by: (None)