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Theorem carduniima 8478
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 5691 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  ->  Fun  F )
2 funimaexg 5621 . . . . 5  |-  ( ( Fun  F  /\  A  e.  B )  ->  ( F " A )  e. 
_V )
31, 2sylan 473 . . . 4  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  A  e.  B )  ->  ( F " A
)  e.  _V )
43expcom 436 . . 3  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  e.  _V )
)
5 ffn 5689 . . . . . . . . 9  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnima 5655 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 17 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  =  ran  F
)
8 frn 5695 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  ->  ran  F  C_  ( om  u.  ran  aleph ) )
97, 8eqsstrd 3441 . . . . . . 7  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  C_  ( om  u.  ran  aleph ) )
109sseld 3406 . . . . . 6  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  x  e.  ( om  u.  ran  aleph ) ) )
11 iscard3 8475 . . . . . 6  |-  ( (
card `  x )  =  x  <->  x  e.  ( om  u.  ran  aleph ) )
1210, 11syl6ibr 230 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  ( card `  x
)  =  x ) )
1312ralrimiv 2777 . . . 4  |-  ( F : A --> ( om  u.  ran  aleph )  ->  A. x  e.  ( F " A ) (
card `  x )  =  x )
14 carduni 8367 . . . 4  |-  ( ( F " A )  e.  _V  ->  ( A. x  e.  ( F " A ) (
card `  x )  =  x  ->  ( card `  U. ( F " A ) )  = 
U. ( F " A ) ) )
1513, 14syl5 33 . . 3  |-  ( ( F " A )  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
164, 15syli 38 . 2  |-  ( A  e.  B  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
17 iscard3 8475 . 2  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
1816, 17syl6ib 229 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 1437    e. wcel 1872   A.wral 2714   _Vcvv 3022    u. cun 3377   U.cuni 4162   ran crn 4797   "cima 4799   Fun wfun 5538    Fn wfn 5539   -->wf 5540   ` cfv 5544   omcom 6650   cardccrd 8321   alephcale 8322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-oi 7978  df-har 8026  df-card 8325  df-aleph 8326
This theorem is referenced by:  cardinfima  8479
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