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Theorem tskurn 9467
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 1077 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ Tarski)
2 simp1r 1078 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → Tr 𝑇)
3 frn 5952 . . . 4 (𝐹:𝐴𝑇 → ran 𝐹𝑇)
433ad2ant3 1076 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
5 tskwe2 9451 . . . . . . 7 (𝑇 ∈ Tarski → 𝑇 ∈ dom card)
61, 5syl 17 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝑇 ∈ dom card)
7 simp2 1054 . . . . . . 7 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
8 trss 4683 . . . . . . 7 (Tr 𝑇 → (𝐴𝑇𝐴𝑇))
92, 7, 8sylc 62 . . . . . 6 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
10 ssnum 8722 . . . . . 6 ((𝑇 ∈ dom card ∧ 𝐴𝑇) → 𝐴 ∈ dom card)
116, 9, 10syl2anc 690 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴 ∈ dom card)
12 ffn 5944 . . . . . . 7 (𝐹:𝐴𝑇𝐹 Fn 𝐴)
13 dffn4 6019 . . . . . . 7 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
1412, 13sylib 206 . . . . . 6 (𝐹:𝐴𝑇𝐹:𝐴onto→ran 𝐹)
15143ad2ant3 1076 . . . . 5 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐹:𝐴onto→ran 𝐹)
16 fodomnum 8740 . . . . 5 (𝐴 ∈ dom card → (𝐹:𝐴onto→ran 𝐹 → ran 𝐹𝐴))
1711, 15, 16sylc 62 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝐴)
18 tsksdom 9434 . . . . 5 ((𝑇 ∈ Tarski ∧ 𝐴𝑇) → 𝐴𝑇)
191, 7, 18syl2anc 690 . . . 4 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → 𝐴𝑇)
20 domsdomtr 7957 . . . 4 ((ran 𝐹𝐴𝐴𝑇) → ran 𝐹𝑇)
2117, 19, 20syl2anc 690 . . 3 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
22 tskssel 9435 . . 3 ((𝑇 ∈ Tarski ∧ ran 𝐹𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
231, 4, 21, 22syl3anc 1317 . 2 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
24 tskuni 9461 . 2 ((𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ ran 𝐹𝑇) → ran 𝐹𝑇)
251, 2, 23, 24syl3anc 1317 1 (((𝑇 ∈ Tarski ∧ Tr 𝑇) ∧ 𝐴𝑇𝐹:𝐴𝑇) → ran 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wcel 1976  wss 3539   cuni 4366   class class class wbr 4577  Tr wtr 4674  dom cdm 5028  ran crn 5029   Fn wfn 5785  wf 5786  ontowfo 5788  cdom 7816  csdm 7817  cardccrd 8621  Tarskictsk 9426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-ac2 9145
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-smo 7307  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-oi 8275  df-har 8323  df-r1 8487  df-card 8625  df-aleph 8626  df-cf 8627  df-acn 8628  df-ac 8799  df-wina 9362  df-ina 9363  df-tsk 9427
This theorem is referenced by:  grutsk1  9499
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