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Theorem cdleme31fv 29846
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
Hypotheses
Ref Expression
cdleme31.o  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
cdleme31.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
cdleme31.c  |-  C  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
Assertion
Ref Expression
cdleme31fv  |-  ( X  e.  B  ->  ( F `  X )  =  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
Distinct variable groups:    x, B    x, C    x,  .<_    x, P    x, Q    x, W    x, s, z, X
Allowed substitution hints:    A( x, z, s)    B( z, s)    C( z, s)    P( z, s)    Q( z, s)    F( x, z, s)    .\/ ( x, z, s)    .<_ ( z, s)    ./\ ( x, z, s)    N( x, z, s)    O( x, z, s)    W( z, s)

Proof of Theorem cdleme31fv
StepHypRef Expression
1 cdleme31.c . . . 4  |-  C  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
2 riotaex 6303 . . . 4  |-  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) )  e.  _V
31, 2eqeltri 2354 . . 3  |-  C  e. 
_V
4 ifexg 3625 . . 3  |-  ( ( C  e.  _V  /\  X  e.  B )  ->  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
53, 4mpan 653 . 2  |-  ( X  e.  B  ->  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )
6 breq1 4027 . . . . . 6  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
76notbid 287 . . . . 5  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
87anbi2d 686 . . . 4  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
9 oveq1 5826 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  ./\  W )  =  ( X  ./\  W ) )
109oveq2d 5835 . . . . . . . . . 10  |-  ( x  =  X  ->  (
s  .\/  ( x  ./\ 
W ) )  =  ( s  .\/  ( X  ./\  W ) ) )
11 id 21 . . . . . . . . . 10  |-  ( x  =  X  ->  x  =  X )
1210, 11eqeq12d 2298 . . . . . . . . 9  |-  ( x  =  X  ->  (
( s  .\/  (
x  ./\  W )
)  =  x  <->  ( s  .\/  ( X  ./\  W
) )  =  X ) )
1312anbi2d 686 . . . . . . . 8  |-  ( x  =  X  ->  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  <-> 
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) ) )
149oveq2d 5835 . . . . . . . . 9  |-  ( x  =  X  ->  ( N  .\/  ( x  ./\  W ) )  =  ( N  .\/  ( X 
./\  W ) ) )
1514eqeq2d 2295 . . . . . . . 8  |-  ( x  =  X  ->  (
z  =  ( N 
.\/  ( x  ./\  W ) )  <->  z  =  ( N  .\/  ( X 
./\  W ) ) ) )
1613, 15imbi12d 313 . . . . . . 7  |-  ( x  =  X  ->  (
( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  -> 
z  =  ( N 
.\/  ( X  ./\  W ) ) ) ) )
1716ralbidv 2564 . . . . . 6  |-  ( x  =  X  ->  ( A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) )  <->  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
1817riotabidv 6301 . . . . 5  |-  ( x  =  X  ->  ( iota_ z  e.  B A. s  e.  A  (
( -.  s  .<_  W  /\  ( s  .\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )  =  (
iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( X  ./\  W ) )  =  X )  ->  z  =  ( N  .\/  ( X 
./\  W ) ) ) ) )
19 cdleme31.o . . . . 5  |-  O  =  ( iota_ z  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( N  .\/  ( x 
./\  W ) ) ) )
2018, 19, 13eqtr4g 2341 . . . 4  |-  ( x  =  X  ->  O  =  C )
218, 20, 11ifbieq12d 3588 . . 3  |-  ( x  =  X  ->  if ( ( P  =/= 
Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
22 cdleme31.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
2321, 22fvmptg 5561 . 2  |-  ( ( X  e.  B  /\  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X )  e.  _V )  ->  ( F `  X )  =  if ( ( P  =/= 
Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
245, 23mpdan 651 1  |-  ( X  e.  B  ->  ( F `  X )  =  if ( ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   _Vcvv 2789   ifcif 3566   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   iota_crio 6290
This theorem is referenced by:  cdleme31fv1  29847
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-iota 6252  df-riota 6299
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