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Theorem carduniima 7607
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
StepHypRef Expression
1 ffun 5248 . . . . 5  |-  ( F : A --> ( om  u.  ran  aleph )  ->  Fun  F )
2 funimaexg 5186 . . . . 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 5246 . . . . . . . . 9  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnima 5219 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
75, 6syl 17 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  =  ran  F
)
8 frn 5252 . . . . . . . 8  |-  ( F : A --> ( om  u.  ran  aleph )  ->  ran  F  C_  ( om  u.  ran  aleph ) )
97, 8eqsstrd 3133 . . . . . . 7  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( F " A
)  C_  ( om  u.  ran  aleph ) )
109sseld 3102 . . . . . 6  |-  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  ( F " A )  ->  x  e.  ( om  u.  ran  aleph ) ) )
11 iscard3 7604 . . . . . 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 2587 . . . 4  |-  ( F : A --> ( om  u.  ran  aleph )  ->  A. x  e.  ( F " A ) (
card `  x )  =  x )
14 carduni 7498 . . . 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 7604 . 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 1619    e. wcel 1621   A.wral 2509   _Vcvv 2727    u. cun 3076   U.cuni 3727   omcom 4547   ran crn 4581   "cima 4583   Fun wfun 4586    Fn wfn 4587   -->wf 4588   ` cfv 4592   cardccrd 7452   alephcale 7453
This theorem is referenced by:  cardinfima  7608
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-har 7156  df-card 7456  df-aleph 7457
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