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Theorem tskurn 9649
 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 1105 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ Tarski)
2 simp1r 1106 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → Tr 𝑇)
3 frn 6091 . . . 4 (𝐹:𝐴𝑇 → ran 𝐹𝑇)
433ad2ant3 1104 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
5 tskwe2 9633 . . . . . . 7 (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
61, 5syl 17 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ dom card)
7 simp2 1082 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
8 trss 4794 . . . . . . 7 (Tr 𝑇 → (𝐴𝑇𝐴𝑇))
92, 7, 8sylc 65 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
10 ssnum 8900 . . . . . 6 ((𝑇 ∈ dom card ∧ 𝐴𝑇) → 𝐴 ∈ dom card)
116, 9, 10syl2anc 694 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴 ∈ dom card)
12 ffn 6083 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
13 dffn4 6159 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1412, 13sylib 208 . . . . . 6 (𝐹:𝐴𝑇𝐹:𝐴onto→ran 𝐹)
15143ad2ant3 1104 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐹:𝐴onto→ran 𝐹)
16 fodomnum 8918 . . . . 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 9616 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
191, 7, 18syl2anc 694 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
20 domsdomtr 8136 . . . 4 ((ran 𝐹𝐴𝐴𝑇) → ran 𝐹𝑇)
2117, 19, 20syl2anc 694 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
22 tskssel 9617 . . 3 ((𝑇 ∈ Tarski ∧ ran 𝐹𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
231, 4, 21, 22syl3anc 1366 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
24 tskuni 9643 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
251, 2, 23, 24syl3anc 1366 1 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   ∈ wcel 2030   ⊆ wss 3607  ∪ cuni 4468   class class class wbr 4685  Tr wtr 4785  dom cdm 5143  ran crn 5144   Fn wfn 5921  ⟶wf 5922  –onto→wfo 5924   ≼ cdom 7995   ≺ csdm 7996  cardccrd 8799  Tarskictsk 9608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-ac2 9323 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-smo 7488  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-har 8504  df-r1 8665  df-card 8803  df-aleph 8804  df-cf 8805  df-acn 8806  df-ac 8977  df-wina 9544  df-ina 9545  df-tsk 9609 This theorem is referenced by:  grutsk1  9681
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