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Theorem fseqdom 8302
Description: One half of fseqen 8303. (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 7955 . . 3  |-  om  e.  _V
2 ovex 6220 . . 3  |-  ( A  ^m  n )  e. 
_V
31, 2iunex 6662 . 2  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
4 xp1st 6711 . . . . . . . 8  |-  ( x  e.  ( om  X.  A )  ->  ( 1st `  x )  e. 
om )
5 peano2 6601 . . . . . . . 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 6712 . . . . . . . . 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 5702 . . . . . . . 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 7332 . . . . . . . 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 6203 . . . . . . . 8  |-  ( n  =  suc  ( 1st `  x )  ->  ( A  ^m  n )  =  ( A  ^m  suc  ( 1st `  x ) ) )
1615eleq2d 2522 . . . . . . 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 3173 . . . . . 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 4278 . . . . 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 4902 . . . . . . 7  |-  suc  ( 1st `  x )  =/=  (/)
23 fvex 5804 . . . . . . . 8  |-  ( 2nd `  x )  e.  _V
2423snnz 4096 . . . . . . 7  |-  { ( 2nd `  x ) }  =/=  (/)
25 xp11 5376 . . . . . . 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 6711 . . . . . . . . 9  |-  ( y  e.  ( om  X.  A )  ->  ( 1st `  y )  e. 
om )
28 peano4 6603 . . . . . . . . 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 4146 . . . . . . . 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 6720 . . . . . 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 7455 . 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 1370    e. wcel 1758    =/= wne 2645   E.wrex 2797   _Vcvv 3072   (/)c0 3740   {csn 3980   U_ciun 4274   class class class wbr 4395   suc csuc 4824    X. cxp 4941   -->wf 5517   ` cfv 5521  (class class class)co 6195   omcom 6581   1stc1st 6680   2ndc2nd 6681    ^m cmap 7319    ~<_ cdom 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-map 7321  df-dom 7417
This theorem is referenced by:  fseqen  8303
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