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Theorem fpwwe2cbv 9001
Description: Lemma for fpwwe2 9014. (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 458 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  x  =  a )
32sseq1d 3429 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  C_  A  <->  a 
C_  A ) )
4 simpr 462 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  r  =  s )
52sqxpeqd 4817 . . . . . 6  |-  ( ( x  =  a  /\  r  =  s )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
64, 5sseq12d 3431 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  C_  (
x  X.  x )  <-> 
s  C_  ( a  X.  a ) ) )
73, 6anbi12d 715 . . . 4  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( x  C_  A  /\  r  C_  (
x  X.  x ) )  <->  ( a  C_  A  /\  s  C_  (
a  X.  a ) ) ) )
8 weeq2 4780 . . . . . 6  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9 weeq1 4779 . . . . . 6  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
108, 9sylan9bb 704 . . . . 5  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  We  x  <->  s  We  a ) )
11 id 22 . . . . . . . . . . 11  |-  ( u  =  v  ->  u  =  v )
1211sqxpeqd 4817 . . . . . . . . . . . 12  |-  ( u  =  v  ->  (
u  X.  u )  =  ( v  X.  v ) )
1312ineq2d 3602 . . . . . . . . . . 11  |-  ( u  =  v  ->  (
r  i^i  ( u  X.  u ) )  =  ( r  i^i  (
v  X.  v ) ) )
1411, 13oveq12d 6262 . . . . . . . . . 10  |-  ( u  =  v  ->  (
u F ( r  i^i  ( u  X.  u ) ) )  =  ( v F ( r  i^i  (
v  X.  v ) ) ) )
1514eqeq1d 2425 . . . . . . . . 9  |-  ( u  =  v  ->  (
( u F ( r  i^i  ( u  X.  u ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  y ) )
1615cbvsbcv 3267 . . . . . . . 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 3946 . . . . . . . . . 10  |-  ( y  =  z  ->  { y }  =  { z } )
1817imaeq2d 5125 . . . . . . . . 9  |-  ( y  =  z  ->  ( `' r " {
y } )  =  ( `' r " { z } ) )
19 eqeq2 2434 . . . . . . . . 9  |-  ( y  =  z  ->  (
( v F ( r  i^i  ( v  X.  v ) ) )  =  y  <->  ( v F ( r  i^i  ( v  X.  v
) ) )  =  z ) )
2018, 19sbceqbid 3244 . . . . . . . 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 ) )
2116, 20syl5bb 260 . . . . . . 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 ) )
2221cbvralv 2991 . . . . . 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 )
234cnveqd 4967 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  `' r  =  `' s )
2423imaeq1d 5124 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( `' r " { z } )  =  ( `' s
" { z } ) )
254ineq1d 3601 . . . . . . . . . 10  |-  ( ( x  =  a  /\  r  =  s )  ->  ( r  i^i  (
v  X.  v ) )  =  ( s  i^i  ( v  X.  v ) ) )
2625oveq2d 6260 . . . . . . . . 9  |-  ( ( x  =  a  /\  r  =  s )  ->  ( v F ( r  i^i  ( v  X.  v ) ) )  =  ( v F ( s  i^i  ( v  X.  v
) ) ) )
2726eqeq1d 2425 . . . . . . . 8  |-  ( ( x  =  a  /\  r  =  s )  ->  ( ( v F ( r  i^i  (
v  X.  v ) ) )  =  z  <-> 
( v F ( s  i^i  ( v  X.  v ) ) )  =  z ) )
2824, 27sbceqbid 3244 . . . . . . 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 ) )
292, 28raleqbidv 2973 . . . . . 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 ) )
3022, 29syl5bb 260 . . . . 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 ) )
3110, 30anbi12d 715 . . . 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 ) ) )
327, 31anbi12d 715 . . 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 ) ) ) )
3332cbvopabv 4431 . 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 ) ) }
341, 33eqtri 2445 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:    /\ wa 370    = wceq 1437   A.wral 2709   [.wsbc 3237    i^i cin 3373    C_ wss 3374   {csn 3936   {copab 4419    We wwe 4749    X. cxp 4789   `'ccnv 4790   "cima 4794  (class class class)co 6244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ral 2714  df-rex 2715  df-rab 2718  df-v 3019  df-sbc 3238  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-br 4362  df-opab 4421  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-cnv 4799  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fv 5547  df-ov 6247
This theorem is referenced by:  fpwwe2lem12  9012  fpwwe2lem13  9013  canthwe  9022  pwfseqlem5  9034
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