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Theorem tskurn 9071
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 1012 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  Tarski )
2 simp1r 1013 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  Tr  T )
3 frn 5676 . . . 4  |-  ( F : A --> T  ->  ran  F  C_  T )
433ad2ant3 1011 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  C_  T
)
5 tskwe2 9055 . . . . . . 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 989 . . . . . . 7  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  T
)
8 trss 4505 . . . . . . 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 8324 . . . . . 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 5670 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
13 dffn4 5737 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1412, 13sylib 196 . . . . . 6  |-  ( F : A --> T  ->  F : A -onto-> ran  F
)
15143ad2ant3 1011 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  F : A -onto-> ran  F )
16 fodomnum 8342 . . . . 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 9038 . . . . 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 7559 . . . 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 9039 . . 3  |-  ( ( T  e.  Tarski  /\  ran  F 
C_  T  /\  ran  F 
~<  T )  ->  ran  F  e.  T )
231, 4, 21, 22syl3anc 1219 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  e.  T )
24 tskuni 9065 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  ran  F  e.  T )  ->  U. ran  F  e.  T )
251, 2, 23, 24syl3anc 1219 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 965    e. wcel 1758    C_ wss 3439   U.cuni 4202   class class class wbr 4403   Tr wtr 4496   dom cdm 4951   ran crn 4952    Fn wfn 5524   -->wf 5525   -onto->wfo 5527    ~<_ cdom 7421    ~< csdm 7422   cardccrd 8220   Tarskictsk 9030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-ac2 8747
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-smo 6920  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7839  df-har 7888  df-r1 8086  df-card 8224  df-aleph 8225  df-cf 8226  df-acn 8227  df-ac 8401  df-wina 8966  df-ina 8967  df-tsk 9031
This theorem is referenced by:  grutsk1  9103
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