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Theorem tskurn 10721
Description: A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013.)
Assertion
Ref Expression
tskurn (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)

Proof of Theorem tskurn
StepHypRef Expression
1 simp1l 1197 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ Tarski)
2 simp1r 1198 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → Tr 𝑇)
3 frn 6672 . . . 4 (𝐹:𝐴𝑇 → ran 𝐹𝑇)
433ad2ant3 1135 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
5 tskwe2 10705 . . . . . . 7 (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
61, 5syl 17 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ dom card)
7 simp2 1137 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
8 trss 5231 . . . . . . 7 (Tr 𝑇 → (𝐴𝑇𝐴𝑇))
92, 7, 8sylc 65 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
10 ssnum 9971 . . . . . 6 ((𝑇 ∈ dom card ∧ 𝐴𝑇) → 𝐴 ∈ dom card)
116, 9, 10syl2anc 584 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴 ∈ dom card)
12 ffn 6665 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
13 dffn4 6759 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1412, 13sylib 217 . . . . . 6 (𝐹:𝐴𝑇𝐹:𝐴onto→ran 𝐹)
15143ad2ant3 1135 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐹:𝐴onto→ran 𝐹)
16 fodomnum 9989 . . . . 5 (𝐴 ∈ dom card → (𝐹:𝐴onto→ran 𝐹 → ran 𝐹𝐴))
1711, 15, 16sylc 65 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝐴)
18 tsksdom 10688 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
191, 7, 18syl2anc 584 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
20 domsdomtr 9052 . . . 4 ((ran 𝐹𝐴𝐴𝑇) → ran 𝐹𝑇)
2117, 19, 20syl2anc 584 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
22 tskssel 10689 . . 3 ((𝑇 ∈ Tarski ∧ ran 𝐹𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
231, 4, 21, 22syl3anc 1371 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
24 tskuni 10715 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
251, 2, 23, 24syl3anc 1371 1 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wcel 2106  wss 3908   cuni 4863   class class class wbr 5103  Tr wtr 5220  dom cdm 5631  ran crn 5632   Fn wfn 6488  wf 6489  ontowfo 6491  cdom 8877  csdm 8878  cardccrd 9867  Tarskictsk 10680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-inf2 9573  ax-ac2 10395
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-frecs 8208  df-wrecs 8239  df-smo 8288  df-recs 8313  df-rdg 8352  df-1o 8408  df-2o 8409  df-er 8644  df-map 8763  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-oi 9442  df-har 9489  df-r1 9696  df-card 9871  df-aleph 9872  df-cf 9873  df-acn 9874  df-ac 10048  df-wina 10616  df-ina 10617  df-tsk 10681
This theorem is referenced by:  grutsk1  10753
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