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Theorem cardinfima 7938
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 2928 . 2  |-  ( A  e.  B  ->  A  e.  _V )
2 isinfcard 7933 . . . . . . . . . . . . 13  |-  ( ( om  C_  ( F `  x )  /\  ( card `  ( F `  x ) )  =  ( F `  x
) )  <->  ( F `  x )  e.  ran  aleph
)
32bicomi 194 . . . . . . . . . . . 12  |-  ( ( F `  x )  e.  ran  aleph  <->  ( om  C_  ( F `  x
)  /\  ( card `  ( F `  x
) )  =  ( F `  x ) ) )
43simplbi 447 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ran  aleph  ->  om  C_  ( F `  x )
)
5 ffn 5554 . . . . . . . . . . . 12  |-  ( F : A --> ( om  u.  ran  aleph )  ->  F  Fn  A )
6 fnfvelrn 5830 . . . . . . . . . . . . . . . 16  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  ran  F
)
76ex 424 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  e.  ran  F
) )
8 fnima 5526 . . . . . . . . . . . . . . . 16  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
98eleq2d 2475 . . . . . . . . . . . . . . 15  |-  ( F  Fn  A  ->  (
( F `  x
)  e.  ( F
" A )  <->  ( F `  x )  e.  ran  F ) )
107, 9sylibrd 226 . . . . . . . . . . . . . 14  |-  ( F  Fn  A  ->  (
x  e.  A  -> 
( F `  x
)  e.  ( F
" A ) ) )
11 elssuni 4007 . . . . . . . . . . . . . 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 419 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  C_  U. ( F " A ) )
145, 13sylan 458 . . . . . . . . . . 11  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  x  e.  A )  ->  ( F `  x
)  C_  U. ( F " A ) )
154, 14sylan9ssr 3326 . . . . . . . . . 10  |-  ( ( ( F : A --> ( om  u.  ran  aleph )  /\  x  e.  A )  /\  ( F `  x
)  e.  ran  aleph )  ->  om  C_  U. ( F
" A ) )
1615anasss 629 . . . . . . . . 9  |-  ( ( F : A --> ( om  u.  ran  aleph )  /\  ( x  e.  A  /\  ( F `  x
)  e.  ran  aleph ) )  ->  om  C_  U. ( F " A ) )
1716a1i 11 . . . . . . . 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 7937 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  ->  U. ( F " A
)  e.  ( om  u.  ran  aleph ) ) )
19 iscard3 7934 . . . . . . . . . 10  |-  ( (
card `  U. ( F
" A ) )  =  U. ( F
" A )  <->  U. ( F " A )  e.  ( om  u.  ran  aleph
) )
2018, 19syl6ibr 219 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( F : A --> ( om  u.  ran  aleph )  -> 
( card `  U. ( F
" A ) )  =  U. ( F
" A ) ) )
2120adantrd 455 . . . . . . . 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 520 . . . . . . 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 7933 . . . . . . 7  |-  ( ( om  C_  U. ( F " A )  /\  ( card `  U. ( F
" A ) )  =  U. ( F
" A ) )  <->  U. ( F " A
)  e.  ran  aleph )
2422, 23syl6ib 218 . . . . . 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 593 . . . . 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 419 . . . 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 2793 . . 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 587 . 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 16 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 359    = wceq 1649    e. wcel 1721   E.wrex 2671   _Vcvv 2920    u. cun 3282    C_ wss 3284   U.cuni 3979   omcom 4808   ran crn 4842   "cima 4844    Fn wfn 5412   -->wf 5413   ` cfv 5417   cardccrd 7782   alephcale 7783
This theorem is referenced by:  alephfplem4  7948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-har 7486  df-card 7786  df-aleph 7787
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