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Theorem carduniima 8364
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 5656 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  ->  Fun  F )
2 funimaexg 5590 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
31, 2sylan 471 . . . 4  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
43expcom 435 . . 3  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  e.  _V )
)
5 ffn 5654 . . . . . . . . 9  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnima 5624 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 16 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  =  ran  F
)
8 frn 5660 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  ->  ran  F  C_  ( om  u.  ran  aleph ) )
97, 8eqsstrd 3485 . . . . . . 7  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  C_  ( om  u.  ran  aleph ) )
109sseld 3450 . . . . . 6  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  x  e.  ( om  u.  ran  aleph ) ) )
11 iscard3 8361 . . . . . 6  |-  ( (
card `  x )  =  x  <->  x  e.  ( om  u.  ran  aleph ) )
1210, 11syl6ibr 227 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  ( card `  x
)  =  x ) )
1312ralrimiv 2818 . . . 4  |-  ( F : A --> ( om  u.  ran  aleph )  ->  A. x  e.  ( F " A ) (
card `  x )  =  x )
14 carduni 8249 . . . 4  |-  ( ( F " A )  e.  _V  ->  ( A. x  e.  ( F " A ) (
card `  x )  =  x  ->  ( card `  U. ( F " A ) )  = 
U. ( F " A ) ) )
1513, 14syl5 32 . . 3  |-  ( ( F " A )  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
164, 15syli 37 . 2  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
17 iscard3 8361 . 2  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
1816, 17syl6ib 226 1  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2793   _Vcvv 3065    u. cun 3421   U.cuni 4186   ran crn 4936   "cima 4938   Fun wfun 5507    Fn wfn 5508   -->wf 5509   ` cfv 5513   omcom 6573   cardccrd 8203   alephcale 8204
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 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945
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 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-om 6574  df-recs 6929  df-rdg 6963  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-oi 7822  df-har 7871  df-card 8207  df-aleph 8208
This theorem is referenced by:  cardinfima  8365
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