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Theorem cover2g 26359
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,  ph." Note that  ph and  x must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
Hypothesis
Ref Expression
cover2g.1  |-  A  = 
U. B
Assertion
Ref Expression
cover2g  |-  ( B  e.  C  ->  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
Distinct variable groups:    ph, x, z   
x, B, y, z   
x, A, z
Allowed substitution hints:    ph( y)    A( y)    C( x, y, z)

Proof of Theorem cover2g
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 unieq 3836 . . . 4  |-  ( b  =  B  ->  U. b  =  U. B )
2 cover2g.1 . . . 4  |-  A  = 
U. B
31, 2syl6eqr 2333 . . 3  |-  ( b  =  B  ->  U. b  =  A )
4 rexeq 2737 . . 3  |-  ( b  =  B  ->  ( E. y  e.  b 
( x  e.  y  /\  ph )  <->  E. y  e.  B  ( x  e.  y  /\  ph )
) )
53, 4raleqbidv 2748 . 2  |-  ( b  =  B  ->  ( A. x  e.  U. b E. y  e.  b 
( x  e.  y  /\  ph )  <->  A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )
) )
6 pweq 3628 . . 3  |-  ( b  =  B  ->  ~P b  =  ~P B
)
73eqeq2d 2294 . . . 4  |-  ( b  =  B  ->  ( U. z  =  U. b 
<-> 
U. z  =  A ) )
87anbi1d 685 . . 3  |-  ( b  =  B  ->  (
( U. z  = 
U. b  /\  A. y  e.  z  ph ) 
<->  ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
96, 8rexeqbidv 2749 . 2  |-  ( b  =  B  ->  ( E. z  e.  ~P  b ( U. z  =  U. b  /\  A. y  e.  z  ph ) 
<->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
10 vex 2791 . . 3  |-  b  e. 
_V
11 eqid 2283 . . 3  |-  U. b  =  U. b
1210, 11cover2 26358 . 2  |-  ( A. x  e.  U. b E. y  e.  b 
( x  e.  y  /\  ph )  <->  E. z  e.  ~P  b ( U. z  =  U. b  /\  A. y  e.  z 
ph ) )
135, 9, 12vtoclbg 2844 1  |-  ( B  e.  C  ->  ( A. x  e.  A  E. y  e.  B  ( x  e.  y  /\  ph )  <->  E. z  e.  ~P  B ( U. z  =  A  /\  A. y  e.  z  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ~Pcpw 3625   U.cuni 3827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828
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