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Theorem fpwwe2cbv 8997
Description: Lemma for fpwwe2 9010. (Contributed by Mario Carneiro, 3-Jun-2015.)
Hypothesis
Ref Expression
fpwwe2.1  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
Assertion
Ref Expression
fpwwe2cbv  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
Distinct variable groups:    y, u    r, a, s, u, v, x, y, z, F    A, a, r, s, x, z
Allowed substitution hints:    A( y, v, u)    W( x, y, z, v, u, s, r, a)

Proof of Theorem fpwwe2cbv
StepHypRef Expression
1 fpwwe2.1 . 2  |-  W  =  { <. x ,  r
>.  |  ( (
x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }
2 simpl 457 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3524 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 461 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52, 2xpeq12d 5017 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3526 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 710 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4861 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4860 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 699 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 id 22 . . . . . . . . . . 11  |-  ( u  =  v  ->  u  =  v )
1211, 11xpeq12d 5017 . . . . . . . . . . . 12  |-  ( u  =  v  ->  (
u  X.  u )  =  ( v  X.  v ) )
1312ineq2d 3693 . . . . . . . . . . 11  |-  ( u  =  v  ->  (
r  i^i  ( u  X.  u ) )  =  ( r  i^i  (
v  X.  v ) ) )
1411, 13oveq12d 6293 . . . . . . . . . 10  |-  ( u  =  v  ->  (
u F ( r  i^i  ( u  X.  u ) ) )  =  ( v F ( r  i^i  (
v  X.  v ) ) ) )
1514eqeq1d 2462 . . . . . . . . 9  |-  ( u  =  v  ->  (
( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  y ) )
1615cbvsbcv 3354 . . . . . . . 8  |-  ( [. ( `' r " {
y } )  /  u ]. ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <->  [. ( `' r " { y } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y )
17 sneq 4030 . . . . . . . . . . 11  |-  ( y  =  z  ->  { y }  =  { z } )
1817imaeq2d 5328 . . . . . . . . . 10  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
19 dfsbcq 3326 . . . . . . . . . 10  |-  ( ( `' r " {
y } )  =  ( `' r " { z } )  ->  ( [. ( `' r " {
y } )  / 
v ]. ( v F ( r  i^i  (
v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y ) )
2018, 19syl 16 . . . . . . . . 9  |-  ( y  =  z  ->  ( [. ( `' r " { y } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  y ) )
21 eqeq2 2475 . . . . . . . . . 10  |-  ( y  =  z  ->  (
( v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  z ) )
2221sbcbidv 3383 . . . . . . . . 9  |-  ( y  =  z  ->  ( [. ( `' r " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2320, 22bitrd 253 . . . . . . . 8  |-  ( y  =  z  ->  ( [. ( `' r " { y } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2416, 23syl5bb 257 . . . . . . 7  |-  ( y  =  z  ->  ( [. ( `' r " { y } )  /  u ]. (
u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2524cbvralv 3081 . . . . . 6  |-  ( A. y  e.  x  [. ( `' r " {
y } )  /  u ]. ( u F ( r  i^i  (
u  X.  u ) ) )  =  y  <->  A. z  e.  x  [. ( `' r " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  z )
264cnveqd 5169 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
2726imaeq1d 5327 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
28 dfsbcq 3326 . . . . . . . . 9  |-  ( ( `' r " {
z } )  =  ( `' s " { z } )  ->  ( [. ( `' r " {
z } )  / 
v ]. ( v F ( r  i^i  (
v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. (
v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
2927, 28syl 16 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z ) )
304ineq1d 3692 . . . . . . . . . . 11  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  i^i  (
v  X.  v ) )  =  ( s  i^i  ( v  X.  v ) ) )
3130oveq2d 6291 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  ( v F ( r  i^i  ( v  X.  v ) ) )  =  ( v F ( s  i^i  ( v  X.  v
) ) ) )
3231eqeq1d 2462 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( v F ( r  i^i  (
v  X.  v ) ) )  =  z  <-> 
( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3332sbcbidv 3383 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( [. ( `' s " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3429, 33bitrd 253 . . . . . . 7  |-  ( ( x  =  a  /\  r  =  s )  ->  ( [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
352, 34raleqbidv 3065 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. z  e.  x  [. ( `' r " { z } )  /  v ]. ( v F ( r  i^i  ( v  X.  v ) ) )  =  z  <->  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3625, 35syl5bb 257 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  A. z  e.  a  [. ( `' s " { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
3710, 36anbi12d 710 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y )  <-> 
( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) )
387, 37anbi12d 710 . . 3  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( ( x 
C_  A  /\  r  C_  ( x  X.  x
) )  /\  (
r  We  x  /\  A. y  e.  x  [. ( `' r " {
y } )  /  u ]. ( u F ( r  i^i  (
u  X.  u ) ) )  =  y ) )  <->  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) ) )
3938cbvopabv 4509 . 2  |-  { <. x ,  r >.  |  ( ( x  C_  A  /\  r  C_  ( x  X.  x ) )  /\  ( r  We  x  /\  A. y  e.  x  [. ( `' r " { y } )  /  u ]. ( u F ( r  i^i  ( u  X.  u ) ) )  =  y ) ) }  =  { <. a ,  s >.  |  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
) )  /\  (
s  We  a  /\  A. z  e.  a  [. ( `' s " {
z } )  / 
v ]. ( v F ( s  i^i  (
v  X.  v ) ) )  =  z ) ) }
401, 39eqtri 2489 1  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. z  e.  a 
[. ( `' s
" { z } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   A.wral 2807   [.wsbc 3324    i^i cin 3468    C_ wss 3469   {csn 4020   {copab 4497    We wwe 4830    X. cxp 4990   `'ccnv 4991   "cima 4995  (class class class)co 6275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fv 5587  df-ov 6278
This theorem is referenced by:  fpwwe2lem12  9008  fpwwe2lem13  9009  canthwe  9018  pwfseqlem5  9030
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