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Theorem carduniima 7719
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 ) ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem carduniima
StepHypRef Expression
1 ffun 5357 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  ->  Fun  F )
2 funimaexg 5295 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
31, 2sylan 459 . . . 4  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
43expcom 426 . . 3  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  e.  _V )
)
5 ffn 5355 . . . . . . . . 9  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnima 5328 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 17 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  =  ran  F
)
8 frn 5361 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  ->  ran  F  C_  ( om  u.  ran  aleph ) )
97, 8eqsstrd 3214 . . . . . . 7  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  C_  ( om  u.  ran  aleph ) )
109sseld 3181 . . . . . 6  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  x  e.  ( om  u.  ran  aleph ) ) )
11 iscard3 7716 . . . . . 6  |-  ( (
card `  x )  =  x  <->  x  e.  ( om  u.  ran  aleph ) )
1210, 11syl6ibr 220 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  ( card `  x
)  =  x ) )
1312ralrimiv 2627 . . . 4  |-  ( F : A --> ( om  u.  ran  aleph )  ->  A. x  e.  ( F " A ) (
card `  x )  =  x )
14 carduni 7610 . . . 4  |-  ( ( F " A )  e.  _V  ->  ( A. x  e.  ( F " A ) (
card `  x )  =  x  ->  ( card `  U. ( F " A ) )  = 
U. ( F " A ) ) )
1513, 14syl5 30 . . 3  |-  ( ( F " A )  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
164, 15syli 35 . 2  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
17 iscard3 7716 . 2  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
1816, 17syl6ib 219 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 6    = wceq 1624    e. wcel 1685   A.wral 2545   _Vcvv 2790    u. cun 3152   U.cuni 3829   omcom 4656   ran crn 4690   "cima 4692   Fun wfun 5216    Fn wfn 5217   -->wf 5218   ` cfv 5222   cardccrd 7564   alephcale 7565
This theorem is referenced by:  cardinfima  7720
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-har 7268  df-card 7568  df-aleph 7569
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