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Theorem tskurn 8664
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 981 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  T  e.  Tarski )
2 simp1r 982 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  Tr  T )
3 frn 5597 . . . 4  |-  ( F : A --> T  ->  ran  F  C_  T )
433ad2ant3 980 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  C_  T
)
5 tskwe2 8648 . . . . . . 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 958 . . . . . . 7  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  T
)
8 trss 4311 . . . . . . 7  |-  ( Tr  T  ->  ( A  e.  T  ->  A  C_  T ) )
92, 7, 8sylc 58 . . . . . 6  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  C_  T
)
10 ssnum 7920 . . . . . 6  |-  ( ( T  e.  dom  card  /\  A  C_  T )  ->  A  e.  dom  card )
116, 9, 10syl2anc 643 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  e.  dom  card )
12 ffn 5591 . . . . . . 7  |-  ( F : A --> T  ->  F  Fn  A )
13 dffn4 5659 . . . . . . 7  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
1412, 13sylib 189 . . . . . 6  |-  ( F : A --> T  ->  F : A -onto-> ran  F
)
15143ad2ant3 980 . . . . 5  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  F : A -onto-> ran  F )
16 fodomnum 7938 . . . . 5  |-  ( A  e.  dom  card  ->  ( F : A -onto-> ran  F  ->  ran  F  ~<_  A ) )
1711, 15, 16sylc 58 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<_  A )
18 tsksdom 8631 . . . . 5  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
191, 7, 18syl2anc 643 . . . 4  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  A  ~<  T )
20 domsdomtr 7242 . . . 4  |-  ( ( ran  F  ~<_  A  /\  A  ~<  T )  ->  ran  F  ~<  T )
2117, 19, 20syl2anc 643 . . 3  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  ~<  T )
22 tskssel 8632 . . 3  |-  ( ( T  e.  Tarski  /\  ran  F 
C_  T  /\  ran  F 
~<  T )  ->  ran  F  e.  T )
231, 4, 21, 22syl3anc 1184 . 2  |-  ( ( ( T  e.  Tarski  /\ 
Tr  T )  /\  A  e.  T  /\  F : A --> T )  ->  ran  F  e.  T )
24 tskuni 8658 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  ran  F  e.  T )  ->  U. ran  F  e.  T )
251, 2, 23, 24syl3anc 1184 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 359    /\ w3a 936    e. wcel 1725    C_ wss 3320   U.cuni 4015   class class class wbr 4212   Tr wtr 4302   dom cdm 4878   ran crn 4879    Fn wfn 5449   -->wf 5450   -onto->wfo 5452    ~<_ cdom 7107    ~< csdm 7108   cardccrd 7822   Tarskictsk 8623
This theorem is referenced by:  grutsk1  8696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-ac2 8343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-smo 6608  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-r1 7690  df-card 7826  df-aleph 7827  df-cf 7828  df-acn 7829  df-ac 7997  df-wina 8559  df-ina 8560  df-tsk 8624
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