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Theorem fpwwecbv 8521
Description: Lemma for fpwwe 8523. (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 445 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3377 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 449 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52, 2xpeq12d 4905 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3379 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 693 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4573 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4572 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 682 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 sneq 3827 . . . . . . . . . 10  |-  ( y  =  z  ->  { y }  =  { z } )
1211imaeq2d 5205 . . . . . . . . 9  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
1312fveq2d 5734 . . . . . . . 8  |-  ( y  =  z  ->  ( F `  ( `' r " { y } ) )  =  ( F `  ( `' r " { z } ) ) )
14 id 21 . . . . . . . 8  |-  ( y  =  z  ->  y  =  z )
1513, 14eqeq12d 2452 . . . . . . 7  |-  ( y  =  z  ->  (
( F `  ( `' r " {
y } ) )  =  y  <->  ( F `  ( `' r " { z } ) )  =  z ) )
1615cbvralv 2934 . . . . . 6  |-  ( A. y  e.  x  ( F `  ( `' r " { y } ) )  =  y  <->  A. z  e.  x  ( F `  ( `' r " { z } ) )  =  z )
174cnveqd 5050 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
1817imaeq1d 5204 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
1918fveq2d 5734 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( F `  ( `' r " {
z } ) )  =  ( F `  ( `' s " {
z } ) ) )
2019eqeq1d 2446 . . . . . . 7  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( F `  ( `' r " {
z } ) )  =  z  <->  ( F `  ( `' s " { z } ) )  =  z ) )
212, 20raleqbidv 2918 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. z  e.  x  ( F `  ( `' r " {
z } ) )  =  z  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2216, 21syl5bb 250 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. y  e.  x  ( F `  ( `' r " {
y } ) )  =  y  <->  A. z  e.  a  ( F `  ( `' s " { z } ) )  =  z ) )
2310, 22anbi12d 693 . . . 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 693 . . 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 4279 . 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 2458 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 360    = wceq 1653   A.wral 2707    C_ wss 3322   {csn 3816   {copab 4267    We wwe 4542    X. cxp 4878   `'ccnv 4879   "cima 4883   ` cfv 5456
This theorem is referenced by:  canthnum  8526  canthp1  8531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fv 5464
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