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Mirrors > Home > MPE Home > Th. List > Mathboxes > cover2g | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cover2g.1 | ⊢ 𝐴 = ∪ 𝐵 |
Ref | Expression |
---|---|
cover2g | ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4880 | . . . 4 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = ∪ 𝐵) | |
2 | cover2g.1 | . . . 4 ⊢ 𝐴 = ∪ 𝐵 | |
3 | 1, 2 | eqtr4di 2791 | . . 3 ⊢ (𝑏 = 𝐵 → ∪ 𝑏 = 𝐴) |
4 | rexeq 3309 | . . 3 ⊢ (𝑏 = 𝐵 → (∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))) | |
5 | 3, 4 | raleqbidv 3318 | . 2 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ ∪ 𝑏∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑))) |
6 | pweq 4578 | . . 3 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵) | |
7 | 3 | eqeq2d 2744 | . . . 4 ⊢ (𝑏 = 𝐵 → (∪ 𝑧 = ∪ 𝑏 ↔ ∪ 𝑧 = 𝐴)) |
8 | 7 | anbi1d 631 | . . 3 ⊢ (𝑏 = 𝐵 → ((∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ (∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))) |
9 | 6, 8 | rexeqbidv 3319 | . 2 ⊢ (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏(∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))) |
10 | vex 3451 | . . 3 ⊢ 𝑏 ∈ V | |
11 | eqid 2733 | . . 3 ⊢ ∪ 𝑏 = ∪ 𝑏 | |
12 | 10, 11 | cover2 36223 | . 2 ⊢ (∀𝑥 ∈ ∪ 𝑏∃𝑦 ∈ 𝑏 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏(∪ 𝑧 = ∪ 𝑏 ∧ ∀𝑦 ∈ 𝑧 𝜑)) |
13 | 5, 9, 12 | vtoclbg 3530 | 1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 𝒫 cpw 4564 ∪ cuni 4869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-in 3921 df-ss 3931 df-pw 4566 df-uni 4870 |
This theorem is referenced by: (None) |
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