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Theorem carduniima 7723
Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
Assertion
Ref Expression
carduniima  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )

Proof of Theorem carduniima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ffun 5391 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  ->  Fun  F )
2 funimaexg 5329 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
31, 2sylan 457 . . . 4  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
43expcom 424 . . 3  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  e.  _V )
)
5 ffn 5389 . . . . . . . . 9  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnima 5362 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 15 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  =  ran  F
)
8 frn 5395 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  ->  ran  F  C_  ( om  u.  ran  aleph ) )
97, 8eqsstrd 3212 . . . . . . 7  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  C_  ( om  u.  ran  aleph ) )
109sseld 3179 . . . . . 6  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  x  e.  ( om  u.  ran  aleph ) ) )
11 iscard3 7720 . . . . . 6  |-  ( (
card `  x )  =  x  <->  x  e.  ( om  u.  ran  aleph ) )
1210, 11syl6ibr 218 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  ( card `  x
)  =  x ) )
1312ralrimiv 2625 . . . 4  |-  ( F : A --> ( om  u.  ran  aleph )  ->  A. x  e.  ( F " A ) (
card `  x )  =  x )
14 carduni 7614 . . . 4  |-  ( ( F " A )  e.  _V  ->  ( A. x  e.  ( F " A ) (
card `  x )  =  x  ->  ( card `  U. ( F " A ) )  = 
U. ( F " A ) ) )
1513, 14syl5 28 . . 3  |-  ( ( F " A )  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
164, 15syli 33 . 2  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
17 iscard3 7720 . 2  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
1816, 17syl6ib 217 1  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150   U.cuni 3827   omcom 4656   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255   cardccrd 7568   alephcale 7569
This theorem is referenced by:  cardinfima  7724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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