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Theorem tskurn 10263
 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 1195 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ Tarski)
2 simp1r 1196 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → Tr 𝑇)
3 frn 6510 . . . 4 (𝐹:𝐴𝑇 → ran 𝐹𝑇)
433ad2ant3 1133 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
5 tskwe2 10247 . . . . . . 7 (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
61, 5syl 17 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ dom card)
7 simp2 1135 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
8 trss 5152 . . . . . . 7 (Tr 𝑇 → (𝐴𝑇𝐴𝑇))
92, 7, 8sylc 65 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
10 ssnum 9513 . . . . . 6 ((𝑇 ∈ dom card ∧ 𝐴𝑇) → 𝐴 ∈ dom card)
116, 9, 10syl2anc 587 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴 ∈ dom card)
12 ffn 6504 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
13 dffn4 6588 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1412, 13sylib 221 . . . . . 6 (𝐹:𝐴𝑇𝐹:𝐴onto→ran 𝐹)
15143ad2ant3 1133 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐹:𝐴onto→ran 𝐹)
16 fodomnum 9531 . . . . 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 10230 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
191, 7, 18syl2anc 587 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
20 domsdomtr 8688 . . . 4 ((ran 𝐹𝐴𝐴𝑇) → ran 𝐹𝑇)
2117, 19, 20syl2anc 587 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
22 tskssel 10231 . . 3 ((𝑇 ∈ Tarski ∧ ran 𝐹𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
231, 4, 21, 22syl3anc 1369 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
24 tskuni 10257 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
251, 2, 23, 24syl3anc 1369 1 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1085   ∈ wcel 2112   ⊆ wss 3861  ∪ cuni 4802   class class class wbr 5037  Tr wtr 5143  dom cdm 5529  ran crn 5530   Fn wfn 6336  ⟶wf 6337  –onto→wfo 6339   ≼ cdom 8539   ≺ csdm 8540  cardccrd 9411  Tarskictsk 10222 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-inf2 9151  ax-ac2 9937 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-iin 4890  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-se 5489  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-isom 6350  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-smo 8000  df-recs 8025  df-rdg 8063  df-1o 8119  df-2o 8120  df-er 8306  df-map 8425  df-ixp 8494  df-en 8542  df-dom 8543  df-sdom 8544  df-fin 8545  df-oi 9021  df-har 9068  df-r1 9240  df-card 9415  df-aleph 9416  df-cf 9417  df-acn 9418  df-ac 9590  df-wina 10158  df-ina 10159  df-tsk 10223 This theorem is referenced by:  grutsk1  10295
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