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Theorem pwfseqlem2 9054
Description: Lemma for pwfseq 9059. (Contributed by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
pwfseqlem4.g  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
pwfseqlem4.x  |-  ( ph  ->  X  C_  A )
pwfseqlem4.h  |-  ( ph  ->  H : om -1-1-onto-> X )
pwfseqlem4.ps  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
pwfseqlem4.k  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
pwfseqlem4.d  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
pwfseqlem4.f  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
Assertion
Ref Expression
pwfseqlem2  |-  ( ( Y  e.  Fin  /\  R  e.  _V )  ->  ( Y F R )  =  ( H `
 ( card `  Y
) ) )
Distinct variable groups:    n, r, w, x, z    D, n, z    w, G    w, K    H, r, x, z    ph, n, r, x, z    ps, n, z    A, n, r, x, z
Allowed substitution hints:    ph( w)    ps( x, w, r)    A( w)    D( x, w, r)    R( x, z, w, n, r)    F( x, z, w, n, r)    G( x, z, n, r)    H( w, n)    K( x, z, n, r)    X( x, z, w, n, r)    Y( x, z, w, n, r)

Proof of Theorem pwfseqlem2
Dummy variables  a 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6303 . . 3  |-  ( a  =  Y  ->  (
a F s )  =  ( Y F s ) )
2 fveq2 5872 . . . 4  |-  ( a  =  Y  ->  ( card `  a )  =  ( card `  Y
) )
32fveq2d 5876 . . 3  |-  ( a  =  Y  ->  ( H `  ( card `  a ) )  =  ( H `  ( card `  Y ) ) )
41, 3eqeq12d 2479 . 2  |-  ( a  =  Y  ->  (
( a F s )  =  ( H `
 ( card `  a
) )  <->  ( Y F s )  =  ( H `  ( card `  Y ) ) ) )
5 oveq2 6304 . . 3  |-  ( s  =  R  ->  ( Y F s )  =  ( Y F R ) )
65eqeq1d 2459 . 2  |-  ( s  =  R  ->  (
( Y F s )  =  ( H `
 ( card `  Y
) )  <->  ( Y F R )  =  ( H `  ( card `  Y ) ) ) )
7 nfcv 2619 . . 3  |-  F/_ x
a
8 nfcv 2619 . . 3  |-  F/_ r
a
9 nfcv 2619 . . 3  |-  F/_ r
s
10 pwfseqlem4.f . . . . . 6  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
11 nfmpt21 6363 . . . . . 6  |-  F/_ x
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
1210, 11nfcxfr 2617 . . . . 5  |-  F/_ x F
13 nfcv 2619 . . . . 5  |-  F/_ x
r
147, 12, 13nfov 6322 . . . 4  |-  F/_ x
( a F r )
1514nfeq1 2634 . . 3  |-  F/ x
( a F r )  =  ( H `
 ( card `  a
) )
16 nfmpt22 6364 . . . . . 6  |-  F/_ r
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
1710, 16nfcxfr 2617 . . . . 5  |-  F/_ r F
188, 17, 9nfov 6322 . . . 4  |-  F/_ r
( a F s )
1918nfeq1 2634 . . 3  |-  F/ r ( a F s )  =  ( H `
 ( card `  a
) )
20 oveq1 6303 . . . 4  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
21 fveq2 5872 . . . . 5  |-  ( x  =  a  ->  ( card `  x )  =  ( card `  a
) )
2221fveq2d 5876 . . . 4  |-  ( x  =  a  ->  ( H `  ( card `  x ) )  =  ( H `  ( card `  a ) ) )
2320, 22eqeq12d 2479 . . 3  |-  ( x  =  a  ->  (
( x F r )  =  ( H `
 ( card `  x
) )  <->  ( a F r )  =  ( H `  ( card `  a ) ) ) )
24 oveq2 6304 . . . 4  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
2524eqeq1d 2459 . . 3  |-  ( r  =  s  ->  (
( a F r )  =  ( H `
 ( card `  a
) )  <->  ( a F s )  =  ( H `  ( card `  a ) ) ) )
26 vex 3112 . . . . . 6  |-  x  e. 
_V
27 vex 3112 . . . . . 6  |-  r  e. 
_V
28 fvex 5882 . . . . . . 7  |-  ( H `
 ( card `  x
) )  e.  _V
29 fvex 5882 . . . . . . 7  |-  ( D `
 |^| { z  e. 
om  |  -.  ( D `  z )  e.  x } )  e. 
_V
3028, 29ifex 4013 . . . . . 6  |-  if ( x  e.  Fin , 
( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )  e. 
_V
3110ovmpt4g 6424 . . . . . 6  |-  ( ( x  e.  _V  /\  r  e.  _V  /\  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )  e. 
_V )  ->  (
x F r )  =  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z
)  e.  x }
) ) )
3226, 27, 30, 31mp3an 1324 . . . . 5  |-  ( x F r )  =  if ( x  e. 
Fin ,  ( H `  ( card `  x
) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )
33 iftrue 3950 . . . . 5  |-  ( x  e.  Fin  ->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) )  =  ( H `  ( card `  x ) ) )
3432, 33syl5eq 2510 . . . 4  |-  ( x  e.  Fin  ->  (
x F r )  =  ( H `  ( card `  x )
) )
3534adantr 465 . . 3  |-  ( ( x  e.  Fin  /\  r  e.  _V )  ->  ( x F r )  =  ( H `
 ( card `  x
) ) )
367, 8, 9, 15, 19, 23, 25, 35vtocl2gaf 3174 . 2  |-  ( ( a  e.  Fin  /\  s  e.  _V )  ->  ( a F s )  =  ( H `
 ( card `  a
) ) )
374, 6, 36vtocl2ga 3175 1  |-  ( ( Y  e.  Fin  /\  R  e.  _V )  ->  ( Y F R )  =  ( H `
 ( card `  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    C_ wss 3471   ifcif 3944   ~Pcpw 4015   |^|cint 4288   U_ciun 4332   class class class wbr 4456    We wwe 4846    X. cxp 5006   `'ccnv 5007   ran crn 5009   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   omcom 6699    ^m cmap 7438    ~<_ cdom 7533   Fincfn 7535   cardccrd 8333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  pwfseqlem4a  9056  pwfseqlem4  9057
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