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Theorem cardinfima 7608
Description: If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
Assertion
Ref Expression
cardinfima  |-  ( A  e.  B  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  E. x  e.  A  ( F `  x )  e.  ran  aleph )  ->  U. ( F " A
)  e.  ran  aleph ) )
Distinct variable groups:    x, F    x, A
Allowed substitution hint:    B( x)

Proof of Theorem cardinfima
StepHypRef Expression
1 elex 2735 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 isinfcard 7603 . . . . . . . . . . . . 13  |-  ( ( om  C_  ( F `  x )  /\  ( card `  ( F `  x ) )  =  ( F `  x
) )  <->  ( F `  x )  e.  ran  aleph
)
32bicomi 195 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  ran  aleph  <->  ( om  C_  ( F `  x
)  /\  ( card `  ( F `  x
) )  =  ( F `  x ) ) )
43simplbi 448 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ran  aleph  ->  om  C_  ( F `  x )
)
5 ffn 5246 . . . . . . . . . . . 12  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnfvelrn 5514 . . . . . . . . . . . . . . . 16  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  ran  F
)
76ex 425 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  e.  ran  F
) )
8 fnima 5219 . . . . . . . . . . . . . . . 16  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
98eleq2d 2320 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  (
( F `  x
)  e.  ( F
" A )  <->  ( F `  x )  e.  ran  F ) )
107, 9sylibrd 227 . . . . . . . . . . . . . 14  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  e.  ( F
" A ) ) )
11 elssuni 3753 . . . . . . . . . . . . . 14  |-  ( ( F `  x )  e.  ( F " A )  ->  ( F `  x )  C_ 
U. ( F " A ) )
1210, 11syl6 31 . . . . . . . . . . . . 13  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  C_  U. ( F " A ) ) )
1312imp 420 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  C_  U. ( F " A ) )
145, 13sylan 459 . . . . . . . . . . 11  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  x  e.  A )  ->  ( F `  x
)  C_  U. ( F " A ) )
154, 14sylan9ssr 3114 . . . . . . . . . 10  |-  ( ( ( F : A --> ( om  u.  ran  aleph )  /\  x  e.  A )  /\  ( F `  x
)  e.  ran  aleph )  ->  om  C_  U. ( F
" A ) )
1615anasss 631 . . . . . . . . 9  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  om  C_  U. ( F " A ) )
1716a1i 12 . . . . . . . 8  |-  ( A  e.  _V  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  om  C_  U. ( F " A ) ) )
18 carduniima 7607 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )
19 iscard3 7604 . . . . . . . . . 10  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
2018, 19syl6ibr 220 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
2120adantrd 456 . . . . . . . 8  |-  ( A  e.  _V  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  ( card `  U. ( F " A ) )  =  U. ( F " A ) ) )
2217, 21jcad 521 . . . . . . 7  |-  ( A  e.  _V  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  ( om  C_  U. ( F " A )  /\  ( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) ) )
23 isinfcard 7603 . . . . . . 7  |-  ( ( om  C_  U. ( F " A )  /\  ( card `  U. ( F
" A ) )  =  U. ( F
" A ) )  <->  U. ( F " A
)  e.  ran  aleph )
2422, 23syl6ib 219 . . . . . 6  |-  ( A  e.  _V  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  U. ( F " A )  e.  ran  aleph
) )
2524exp4d 595 . . . . 5  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( x  e.  A  ->  ( ( F `  x )  e.  ran  aleph  ->  U. ( F " A )  e.  ran  aleph
) ) ) )
2625imp 420 . . . 4  |-  ( ( A  e.  _V  /\  F : A --> ( om  u.  ran  aleph ) )  ->  ( x  e.  A  ->  ( ( F `  x )  e.  ran  aleph  ->  U. ( F " A )  e. 
ran  aleph ) ) )
2726rexlimdv 2628 . . 3  |-  ( ( A  e.  _V  /\  F : A --> ( om  u.  ran  aleph ) )  ->  ( E. x  e.  A  ( F `  x )  e.  ran  aleph  ->  U. ( F " A )  e.  ran  aleph
) )
2827expimpd 589 . 2  |-  ( A  e.  _V  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  E. x  e.  A  ( F `  x )  e.  ran  aleph )  ->  U. ( F " A
)  e.  ran  aleph ) )
291, 28syl 17 1  |-  ( A  e.  B  ->  (
( F : A --> ( om  u.  ran  aleph )  /\  E. x  e.  A  ( F `  x )  e.  ran  aleph )  ->  U. ( F " A
)  e.  ran  aleph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2510   _Vcvv 2727    u. cun 3076    C_ wss 3078   U.cuni 3727   omcom 4547   ran crn 4581   "cima 4583    Fn wfn 4587   -->wf 4588   ` cfv 4592   cardccrd 7452   alephcale 7453
This theorem is referenced by:  alephfplem4  7618
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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