Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cover2g Structured version   Visualization version   GIF version

Theorem cover2g 33127
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 (𝐵𝐶 → (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 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.
StepHypRef Expression
1 unieq 4415 . . . 4 (𝑏 = 𝐵 𝑏 = 𝐵)
2 cover2g.1 . . . 4 𝐴 = 𝐵
31, 2syl6eqr 2678 . . 3 (𝑏 = 𝐵 𝑏 = 𝐴)
4 rexeq 3133 . . 3 (𝑏 = 𝐵 → (∃𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑦𝐵 (𝑥𝑦𝜑)))
53, 4raleqbidv 3146 . 2 (𝑏 = 𝐵 → (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)))
6 pweq 4138 . . 3 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
73eqeq2d 2636 . . . 4 (𝑏 = 𝐵 → ( 𝑧 = 𝑏 𝑧 = 𝐴))
87anbi1d 740 . . 3 (𝑏 = 𝐵 → (( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
96, 8rexeqbidv 3147 . 2 (𝑏 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
10 vex 3194 . . 3 𝑏 ∈ V
11 eqid 2626 . . 3 𝑏 = 𝑏
1210, 11cover2 33126 . 2 (∀𝑥 𝑏𝑦𝑏 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝑏( 𝑧 = 𝑏 ∧ ∀𝑦𝑧 𝜑))
135, 9, 12vtoclbg 3258 1 (𝐵𝐶 → (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  𝒫 cpw 4135   cuni 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-in 3567  df-ss 3574  df-pw 4137  df-uni 4408
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator