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Theorem fpwwecbv 8282
Description: Lemma for fpwwe 8284. (Contributed by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
fpwwe.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
Assertion
Ref Expression
fpwwecbv  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) }
Distinct variable groups:    r, a,
s, x, A    y,
a, z, F, r, s, x
Allowed substitution hints:    A( y, z)    W( x, y, z, s, r, a)

Proof of Theorem fpwwecbv
StepHypRef Expression
1 fpwwe.1 . 2  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }
2 simpl 443 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3218 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 447 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52, 2xpeq12d 4730 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3220 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 691 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4398 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4397 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 680 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 sneq 3664 . . . . . . . . . 10  |-  ( y  =  z  ->  { y }  =  { z } )
1211imaeq2d 5028 . . . . . . . . 9  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
1312fveq2d 5545 . . . . . . . 8  |-  ( y  =  z  ->  ( F `  ( `' r " { y } ) )  =  ( F `  ( `' r " { z } ) ) )
14 id 19 . . . . . . . 8  |-  ( y  =  z  ->  y  =  z )
1513, 14eqeq12d 2310 . . . . . . 7  |-  ( y  =  z  ->  (
( F `  ( `' r " {
y } ) )  =  y  <->  ( F `  ( `' r " { z } ) )  =  z ) )
1615cbvralv 2777 . . . . . 6  |-  ( A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y  <->  A. z  e.  x  ( F `  ( `' r " { z } ) )  =  z )
174cnveqd 4873 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
1817imaeq1d 5027 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
1918fveq2d 5545 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( F `  ( `' r " {
z } ) )  =  ( F `  ( `' s " {
z } ) ) )
2019eqeq1d 2304 . . . . . . 7  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( F `  ( `' r " {
z } ) )  =  z  <->  ( F `  ( `' s " { z } ) )  =  z ) )
212, 20raleqbidv 2761 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. z  e.  x  ( F `  ( `' r " {
z } ) )  =  z  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2216, 21syl5bb 248 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. y  e.  x  ( F `  ( `' r " {
y } ) )  =  y  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2310, 22anbi12d 691 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y )  <-> 
( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) )
247, 23anbi12d 691 . . 3  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) )  <->  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) ) )
2524cbvopabv 4104 . 2  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) ) }
261, 25eqtri 2316 1  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a  ( F `  ( `' s " {
z } ) )  =  z ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   A.wral 2556    C_ wss 3165   {csn 3653   {copab 4092    We wwe 4367    X. cxp 4703   `'ccnv 4704   "cima 4708   ` cfv 5271
This theorem is referenced by:  canthnum  8287  canthp1  8292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279
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