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Theorem cardinfima 7726
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 2798 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 isinfcard 7721 . . . . . . . . . . . . 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 5391 . . . . . . . . . . . 12  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnfvelrn 5664 . . . . . . . . . . . . . . . 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 5364 . . . . . . . . . . . . . . . 16  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
98eleq2d 2352 . . . . . . . . . . . . . . 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 3857 . . . . . . . . . . . . . 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 3195 . . . . . . . . . 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 7725 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )
19 iscard3 7722 . . . . . . . . . 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 7721 . . . . . . 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 2668 . . 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 1625    e. wcel 1686   E.wrex 2546   _Vcvv 2790    u. cun 3152    C_ wss 3154   U.cuni 3829   omcom 4658   ran crn 4692   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257   cardccrd 7570   alephcale 7571
This theorem is referenced by:  alephfplem4  7736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-har 7274  df-card 7574  df-aleph 7575
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