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Theorem iundom 8809
Description: An upper bound for the cardinality of an indexed union.  C depends on  x and should be thought of as  C ( x ). (Contributed by NM, 26-Mar-2006.)
Assertion
Ref Expression
iundom  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  C  ~<_  ( A  X.  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    C( x)    V( x)

Proof of Theorem iundom
StepHypRef Expression
1 eqid 2451 . 2  |-  U_ x  e.  A  ( {
x }  X.  C
)  =  U_ x  e.  A  ( {
x }  X.  C
)
2 simpl 457 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  A  e.  V )
3 ovex 6217 . . . . . 6  |-  ( B  ^m  C )  e. 
_V
43rgenw 2893 . . . . 5  |-  A. x  e.  A  ( B  ^m  C )  e.  _V
5 iunexg 6655 . . . . 5  |-  ( ( A  e.  V  /\  A. x  e.  A  ( B  ^m  C )  e.  _V )  ->  U_ x  e.  A  ( B  ^m  C )  e.  _V )
62, 4, 5sylancl 662 . . . 4  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  ( B  ^m  C )  e.  _V )
7 numth3 8742 . . . 4  |-  ( U_ x  e.  A  ( B  ^m  C )  e. 
_V  ->  U_ x  e.  A  ( B  ^m  C )  e.  dom  card )
86, 7syl 16 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  ( B  ^m  C )  e.  dom  card )
9 numacn 8322 . . 3  |-  ( A  e.  V  ->  ( U_ x  e.  A  ( B  ^m  C )  e.  dom  card  ->  U_ x  e.  A  ( B  ^m  C )  e. AC  A ) )
102, 8, 9sylc 60 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  ( B  ^m  C )  e. AC  A )
11 simpr 461 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  A. x  e.  A  C  ~<_  B )
12 reldom 7418 . . . . . 6  |-  Rel  ~<_
1312brrelexi 4979 . . . . 5  |-  ( C  ~<_  B  ->  C  e.  _V )
1413ralimi 2811 . . . 4  |-  ( A. x  e.  A  C  ~<_  B  ->  A. x  e.  A  C  e.  _V )
15 iunexg 6655 . . . 4  |-  ( ( A  e.  V  /\  A. x  e.  A  C  e.  _V )  ->  U_ x  e.  A  C  e.  _V )
1614, 15sylan2 474 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  C  e.  _V )
171, 10, 11iundom2g 8807 . . . 4  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  ( {
x }  X.  C
)  ~<_  ( A  X.  B ) )
1812brrelex2i 4980 . . . 4  |-  ( U_ x  e.  A  ( { x }  X.  C )  ~<_  ( A  X.  B )  -> 
( A  X.  B
)  e.  _V )
19 numth3 8742 . . . 4  |-  ( ( A  X.  B )  e.  _V  ->  ( A  X.  B )  e. 
dom  card )
2017, 18, 193syl 20 . . 3  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  ( A  X.  B )  e. 
dom  card )
21 numacn 8322 . . 3  |-  ( U_ x  e.  A  C  e.  _V  ->  ( ( A  X.  B )  e. 
dom  card  ->  ( A  X.  B )  e. AC  U_ x  e.  A  C )
)
2216, 20, 21sylc 60 . 2  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  ( A  X.  B )  e. AC  U_ x  e.  A  C
)
231, 10, 11, 22iundomg 8808 1  |-  ( ( A  e.  V  /\  A. x  e.  A  C  ~<_  B )  ->  U_ x  e.  A  C  ~<_  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   A.wral 2795   _Vcvv 3070   {csn 3977   U_ciun 4271   class class class wbr 4392    X. cxp 4938   dom cdm 4940  (class class class)co 6192    ^m cmap 7316    ~<_ cdom 7410   cardccrd 8208  AC wacn 8211
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-ac2 8735
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-recs 6934  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-card 8212  df-acn 8215  df-ac 8389
This theorem is referenced by:  unidom  8810  alephreg  8849  inar1  9045
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