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Theorem tskurn 9179
Description: A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
tskurn  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  U. ran  F  e.  T )

Proof of Theorem tskurn
StepHypRef Expression
1 simp1l 1020 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  Tarski )
2 simp1r 1021 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  Tr  T )
3 frn 5743 . . . 4  |-  ( F : A --> T  ->  ran  F  C_  T )
433ad2ant3 1019 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  C_  T
)
5 tskwe2 9163 . . . . . . 7  |-  ( T  e.  Tarski  ->  T  e.  dom  card )
61, 5syl 16 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  dom  card )
7 simp2 997 . . . . . . 7  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  T
)
8 trss 4555 . . . . . . 7  |-  ( Tr  T  ->  ( A  e.  T  ->  A  C_  T ) )
92, 7, 8sylc 60 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  C_  T
)
10 ssnum 8432 . . . . . 6  |-  ( ( T  e.  dom  card  /\  A  C_  T )  ->  A  e.  dom  card )
116, 9, 10syl2anc 661 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  dom  card )
12 ffn 5737 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
13 dffn4 5807 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1412, 13sylib 196 . . . . . 6  |-  ( F : A --> T  ->  F : A -onto-> ran  F
)
15143ad2ant3 1019 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  F : A -onto-> ran  F )
16 fodomnum 8450 . . . . 5  |-  ( A  e.  dom  card  ->  ( F : A -onto-> ran  F  ->  ran  F  ~<_  A ) )
1711, 15, 16sylc 60 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<_  A )
18 tsksdom 9146 . . . . 5  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
191, 7, 18syl2anc 661 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  ~<  T )
20 domsdomtr 7664 . . . 4  |-  ( ( ran  F  ~<_  A  /\  A  ~<  T )  ->  ran  F  ~<  T )
2117, 19, 20syl2anc 661 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<  T )
22 tskssel 9147 . . 3  |-  ( ( T  e.  Tarski  /\  ran  F 
C_  T  /\  ran  F 
~<  T )  ->  ran  F  e.  T )
231, 4, 21, 22syl3anc 1228 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  e.  T )
24 tskuni 9173 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  ran  F  e.  T )  ->  U. ran  F  e.  T )
251, 2, 23, 24syl3anc 1228 1  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  U. ran  F  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767    C_ wss 3481   U.cuni 4251   class class class wbr 4453   Tr wtr 4546   dom cdm 5005   ran crn 5006    Fn wfn 5589   -->wf 5590   -onto->wfo 5592    ~<_ cdom 7526    ~< csdm 7527   cardccrd 8328   Tarskictsk 9138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-ac2 8855
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-smo 7029  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-oi 7947  df-har 7996  df-r1 8194  df-card 8332  df-aleph 8333  df-cf 8334  df-acn 8335  df-ac 8509  df-wina 9074  df-ina 9075  df-tsk 9139
This theorem is referenced by:  grutsk1  9211
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