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Theorem fseqdom 8407
Description: One half of fseqen 8408. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
fseqdom  |-  ( A  e.  V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
Distinct variable group:    A, n
Allowed substitution hint:    V( n)

Proof of Theorem fseqdom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 8060 . . 3  |-  om  e.  _V
2 ovex 6309 . . 3  |-  ( A  ^m  n )  e. 
_V
31, 2iunex 6764 . 2  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
4 xp1st 6814 . . . . . . . 8  |-  ( x  e.  ( om  X.  A )  ->  ( 1st `  x )  e. 
om )
5 peano2 6704 . . . . . . . 8  |-  ( ( 1st `  x )  e.  om  ->  suc  ( 1st `  x )  e.  om )
64, 5syl 16 . . . . . . 7  |-  ( x  e.  ( om  X.  A )  ->  suc  ( 1st `  x )  e.  om )
76adantl 466 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  suc  ( 1st `  x
)  e.  om )
8 xp2nd 6815 . . . . . . . . 9  |-  ( x  e.  ( om  X.  A )  ->  ( 2nd `  x )  e.  A )
98adantl 466 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( 2nd `  x
)  e.  A )
10 fconst6g 5774 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  A  ->  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } ) : suc  ( 1st `  x ) --> A )
119, 10syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A )
12 elmapg 7433 . . . . . . . 8  |-  ( ( A  e.  V  /\  suc  ( 1st `  x
)  e.  om )  ->  ( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) )  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A ) )
136, 12sylan2 474 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) )  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A ) )
1411, 13mpbird 232 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )
15 oveq2 6292 . . . . . . . 8  |-  ( n  =  suc  ( 1st `  x )  ->  ( A  ^m  n )  =  ( A  ^m  suc  ( 1st `  x ) ) )
1615eleq2d 2537 . . . . . . 7  |-  ( n  =  suc  ( 1st `  x )  ->  (
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  n
)  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) ) )
1716rspcev 3214 . . . . . 6  |-  ( ( suc  ( 1st `  x
)  e.  om  /\  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )  ->  E. n  e.  om  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  n
) )
187, 14, 17syl2anc 661 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  E. n  e.  om  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  ( A  ^m  n ) )
19 eliun 4330 . . . . 5  |-  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  U_ n  e.  om  ( A  ^m  n )  <->  E. n  e.  om  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  n
) )
2018, 19sylibr 212 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) )
2120ex 434 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( om 
X.  A )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) ) )
22 nsuceq0 4958 . . . . . . 7  |-  suc  ( 1st `  x )  =/=  (/)
23 fvex 5876 . . . . . . . 8  |-  ( 2nd `  x )  e.  _V
2423snnz 4145 . . . . . . 7  |-  { ( 2nd `  x ) }  =/=  (/)
25 xp11 5442 . . . . . . 7  |-  ( ( suc  ( 1st `  x
)  =/=  (/)  /\  {
( 2nd `  x
) }  =/=  (/) )  -> 
( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  =  ( suc  ( 1st `  y )  X.  {
( 2nd `  y
) } )  <->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y )  /\  {
( 2nd `  x
) }  =  {
( 2nd `  y
) } ) ) )
2622, 24, 25mp2an 672 . . . . . 6  |-  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y )  /\  {
( 2nd `  x
) }  =  {
( 2nd `  y
) } ) )
27 xp1st 6814 . . . . . . . . 9  |-  ( y  e.  ( om  X.  A )  ->  ( 1st `  y )  e. 
om )
28 peano4 6706 . . . . . . . . 9  |-  ( ( ( 1st `  x
)  e.  om  /\  ( 1st `  y )  e.  om )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
294, 27, 28syl2an 477 . . . . . . . 8  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
3029adantl 466 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y
)  <->  ( 1st `  x
)  =  ( 1st `  y ) ) )
31 sneqbg 4197 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  _V  ->  ( { ( 2nd `  x
) }  =  {
( 2nd `  y
) }  <->  ( 2nd `  x )  =  ( 2nd `  y ) ) )
3223, 31mp1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( { ( 2nd `  x ) }  =  { ( 2nd `  y ) }  <->  ( 2nd `  x
)  =  ( 2nd `  y ) ) )
3330, 32anbi12d 710 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  =  suc  ( 1st `  y )  /\  { ( 2nd `  x
) }  =  {
( 2nd `  y
) } )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) ) ) )
3426, 33syl5bb 257 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) ) ) )
35 xpopth 6823 . . . . . 6  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( ( ( 1st `  x )  =  ( 1st `  y )  /\  ( 2nd `  x
)  =  ( 2nd `  y ) )  <->  x  =  y ) )
3635adantl 466 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) )  <->  x  =  y
) )
3734, 36bitrd 253 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  x  =  y ) )
3837ex 434 . . 3  |-  ( A  e.  V  ->  (
( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  x  =  y ) ) )
3921, 38dom2d 7556 . 2  |-  ( A  e.  V  ->  ( U_ n  e.  om  ( A  ^m  n
)  e.  _V  ->  ( om  X.  A )  ~<_ 
U_ n  e.  om  ( A  ^m  n
) ) )
403, 39mpi 17 1  |-  ( A  e.  V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113   (/)c0 3785   {csn 4027   U_ciun 4325   class class class wbr 4447   suc csuc 4880    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6284   omcom 6684   1stc1st 6782   2ndc2nd 6783    ^m cmap 7420    ~<_ cdom 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-map 7422  df-dom 7518
This theorem is referenced by:  fseqen  8408
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