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Theorem tskurn 8411
Description: A transitive Tarski's 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 979 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  Tarski )
2 simp1r 980 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  Tr  T )
3 frn 5395 . . . 4  |-  ( F : A --> T  ->  ran  F  C_  T )
433ad2ant3 978 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  C_  T
)
5 tskwe2 8395 . . . . . . 7  |-  ( T  e.  Tarski  ->  T  e.  dom  card )
61, 5syl 15 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  dom  card )
7 simp2 956 . . . . . . 7  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  T
)
8 trss 4122 . . . . . . 7  |-  ( Tr  T  ->  ( A  e.  T  ->  A  C_  T ) )
92, 7, 8sylc 56 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  C_  T
)
10 ssnum 7666 . . . . . 6  |-  ( ( T  e.  dom  card  /\  A  C_  T )  ->  A  e.  dom  card )
116, 9, 10syl2anc 642 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  dom  card )
12 ffn 5389 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
13 dffn4 5457 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1412, 13sylib 188 . . . . . 6  |-  ( F : A --> T  ->  F : A -onto-> ran  F
)
15143ad2ant3 978 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  F : A -onto-> ran  F )
16 fodomnum 7684 . . . . 5  |-  ( A  e.  dom  card  ->  ( F : A -onto-> ran  F  ->  ran  F  ~<_  A ) )
1711, 15, 16sylc 56 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<_  A )
18 tsksdom 8378 . . . . 5  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
191, 7, 18syl2anc 642 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  ~<  T )
20 domsdomtr 6996 . . . 4  |-  ( ( ran  F  ~<_  A  /\  A  ~<  T )  ->  ran  F  ~<  T )
2117, 19, 20syl2anc 642 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<  T )
22 tskssel 8379 . . 3  |-  ( ( T  e.  Tarski  /\  ran  F 
C_  T  /\  ran  F 
~<  T )  ->  ran  F  e.  T )
231, 4, 21, 22syl3anc 1182 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  e.  T )
24 tskuni 8405 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  ran  F  e.  T )  ->  U. ran  F  e.  T )
251, 2, 23, 24syl3anc 1182 1  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  U. ran  F  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   U.cuni 3827   class class class wbr 4023   Tr wtr 4113   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   -onto->wfo 5253    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568   Tarskictsk 8370
This theorem is referenced by:  grutsk1  8443
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-ac2 8089
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-smo 6363  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-r1 7436  df-card 7572  df-aleph 7573  df-cf 7574  df-acn 7575  df-ac 7743  df-wina 8306  df-ina 8307  df-tsk 8371
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