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Theorem tskurn 9232
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 1054 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  Tarski )
2 simp1r 1055 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  Tr  T )
3 frn 5747 . . . 4  |-  ( F : A --> T  ->  ran  F  C_  T )
433ad2ant3 1053 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  C_  T
)
5 tskwe2 9216 . . . . . . 7  |-  ( T  e.  Tarski  ->  T  e.  dom  card )
61, 5syl 17 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  dom  card )
7 simp2 1031 . . . . . . 7  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  T
)
8 trss 4499 . . . . . . 7  |-  ( Tr  T  ->  ( A  e.  T  ->  A  C_  T ) )
92, 7, 8sylc 61 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  C_  T
)
10 ssnum 8488 . . . . . 6  |-  ( ( T  e.  dom  card  /\  A  C_  T )  ->  A  e.  dom  card )
116, 9, 10syl2anc 673 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  dom  card )
12 ffn 5739 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
13 dffn4 5812 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1412, 13sylib 201 . . . . . 6  |-  ( F : A --> T  ->  F : A -onto-> ran  F
)
15143ad2ant3 1053 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  F : A -onto-> ran  F )
16 fodomnum 8506 . . . . 5  |-  ( A  e.  dom  card  ->  ( F : A -onto-> ran  F  ->  ran  F  ~<_  A ) )
1711, 15, 16sylc 61 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<_  A )
18 tsksdom 9199 . . . . 5  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
191, 7, 18syl2anc 673 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  ~<  T )
20 domsdomtr 7725 . . . 4  |-  ( ( ran  F  ~<_  A  /\  A  ~<  T )  ->  ran  F  ~<  T )
2117, 19, 20syl2anc 673 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<  T )
22 tskssel 9200 . . 3  |-  ( ( T  e.  Tarski  /\  ran  F 
C_  T  /\  ran  F 
~<  T )  ->  ran  F  e.  T )
231, 4, 21, 22syl3anc 1292 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  e.  T )
24 tskuni 9226 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  ran  F  e.  T )  ->  U. ran  F  e.  T )
251, 2, 23, 24syl3anc 1292 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 376    /\ w3a 1007    e. wcel 1904    C_ wss 3390   U.cuni 4190   class class class wbr 4395   Tr wtr 4490   dom cdm 4839   ran crn 4840    Fn wfn 5584   -->wf 5585   -onto->wfo 5587    ~<_ cdom 7585    ~< csdm 7586   cardccrd 8387   Tarskictsk 9191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-ac2 8911
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-smo 7083  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-har 8091  df-r1 8253  df-card 8391  df-aleph 8392  df-cf 8393  df-acn 8394  df-ac 8565  df-wina 9127  df-ina 9128  df-tsk 9192
This theorem is referenced by:  grutsk1  9264
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