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Theorem tskurn 9795
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 1237 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ Tarski)
2 simp1r 1238 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → Tr 𝑇)
3 frn 6206 . . . 4 (𝐹:𝐴𝑇 → ran 𝐹𝑇)
433ad2ant3 1129 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
5 tskwe2 9779 . . . . . . 7 (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
61, 5syl 17 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ dom card)
7 simp2 1131 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
8 trss 4905 . . . . . . 7 (Tr 𝑇 → (𝐴𝑇𝐴𝑇))
92, 7, 8sylc 65 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
10 ssnum 9044 . . . . . 6 ((𝑇 ∈ dom card ∧ 𝐴𝑇) → 𝐴 ∈ dom card)
116, 9, 10syl2anc 696 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴 ∈ dom card)
12 ffn 6198 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
13 dffn4 6274 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1412, 13sylib 208 . . . . . 6 (𝐹:𝐴𝑇𝐹:𝐴onto→ran 𝐹)
15143ad2ant3 1129 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐹:𝐴onto→ran 𝐹)
16 fodomnum 9062 . . . . 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 9762 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
191, 7, 18syl2anc 696 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
20 domsdomtr 8252 . . . 4 ((ran 𝐹𝐴𝐴𝑇) → ran 𝐹𝑇)
2117, 19, 20syl2anc 696 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
22 tskssel 9763 . . 3 ((𝑇 ∈ Tarski ∧ ran 𝐹𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
231, 4, 21, 22syl3anc 1473 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
24 tskuni 9789 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
251, 2, 23, 24syl3anc 1473 1 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2131  wss 3707   cuni 4580   class class class wbr 4796  Tr wtr 4896  dom cdm 5258  ran crn 5259   Fn wfn 6036  wf 6037  ontowfo 6039  cdom 8111  csdm 8112  cardccrd 8943  Tarskictsk 9754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-ac2 9469
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-smo 7604  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-oi 8572  df-har 8620  df-r1 8792  df-card 8947  df-aleph 8948  df-cf 8949  df-acn 8950  df-ac 9121  df-wina 9690  df-ina 9691  df-tsk 9755
This theorem is referenced by:  grutsk1  9827
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