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Theorem cardinfima 7720
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 2797 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 isinfcard 7715 . . . . . . . . . . . . 13  |-  ( ( om  C_  ( F `  x )  /\  ( card `  ( F `  x ) )  =  ( F `  x
) )  <->  ( F `  x )  e.  ran  aleph
)
32bicomi 193 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  ran  aleph  <->  ( om  C_  ( F `  x
)  /\  ( card `  ( F `  x
) )  =  ( F `  x ) ) )
43simplbi 446 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ran  aleph  ->  om  C_  ( F `  x )
)
5 ffn 5355 . . . . . . . . . . . 12  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnfvelrn 5624 . . . . . . . . . . . . . . . 16  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  ran  F
)
76ex 423 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  e.  ran  F
) )
8 fnima 5328 . . . . . . . . . . . . . . . 16  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
98eleq2d 2351 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  (
( F `  x
)  e.  ( F
" A )  <->  ( F `  x )  e.  ran  F ) )
107, 9sylibrd 225 . . . . . . . . . . . . . 14  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  e.  ( F
" A ) ) )
11 elssuni 3856 . . . . . . . . . . . . . 14  |-  ( ( F `  x )  e.  ( F " A )  ->  ( F `  x )  C_ 
U. ( F " A ) )
1210, 11syl6 29 . . . . . . . . . . . . 13  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  C_  U. ( F " A ) ) )
1312imp 418 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  C_  U. ( F " A ) )
145, 13sylan 457 . . . . . . . . . . 11  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  x  e.  A )  ->  ( F `  x
)  C_  U. ( F " A ) )
154, 14sylan9ssr 3194 . . . . . . . . . 10  |-  ( ( ( F : A --> ( om  u.  ran  aleph )  /\  x  e.  A )  /\  ( F `  x
)  e.  ran  aleph )  ->  om  C_  U. ( F
" A ) )
1615anasss 628 . . . . . . . . 9  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  om  C_  U. ( F " A ) )
1716a1i 10 . . . . . . . 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 7719 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )
19 iscard3 7716 . . . . . . . . . 10  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
2018, 19syl6ibr 218 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
2120adantrd 454 . . . . . . . 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 519 . . . . . . 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 7715 . . . . . . 7  |-  ( ( om  C_  U. ( F " A )  /\  ( card `  U. ( F
" A ) )  =  U. ( F
" A ) )  <->  U. ( F " A
)  e.  ran  aleph )
2422, 23syl6ib 217 . . . . . 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 592 . . . . 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 418 . . . 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 2667 . . 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 586 . 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 15 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 4    /\ wa 358    = wceq 1623    e. wcel 1685   E.wrex 2545   _Vcvv 2789    u. cun 3151    C_ wss 3153   U.cuni 3828   omcom 4655   ran crn 4689   "cima 4691    Fn wfn 5216   -->wf 5217   ` cfv 5221   cardccrd 7564   alephcale 7565
This theorem is referenced by:  alephfplem4  7730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  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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