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Theorem itunitc1 10414
Description: Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Hypothesis
Ref Expression
ituni.u 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
Assertion
Ref Expression
itunitc1 ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunitc1
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
21fveq1d 6893 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎)‘𝐵) = ((𝑈𝐴)‘𝐵))
3 fveq2 6891 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
42, 3sseq12d 4015 . . 3 (𝑎 = 𝐴 → (((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5 fveq2 6891 . . . . . 6 (𝑏 = ∅ → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘∅))
65sseq1d 4013 . . . . 5 (𝑏 = ∅ → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)))
7 fveq2 6891 . . . . . 6 (𝑏 = 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝑐))
87sseq1d 4013 . . . . 5 (𝑏 = 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9 fveq2 6891 . . . . . 6 (𝑏 = suc 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘suc 𝑐))
109sseq1d 4013 . . . . 5 (𝑏 = suc 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11 fveq2 6891 . . . . . 6 (𝑏 = 𝐵 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝐵))
1211sseq1d 4013 . . . . 5 (𝑏 = 𝐵 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎)))
13 ituni.u . . . . . . . 8 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
1413ituni0 10412 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
15 tcid 9733 . . . . . . 7 (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
1614, 15eqsstrd 4020 . . . . . 6 (𝑎 ∈ V → ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎))
1716elv 3480 . . . . 5 ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)
1813itunisuc 10413 . . . . . . 7 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
19 tctr 9734 . . . . . . . . . 10 Tr (TC‘𝑎)
20 pwtr 5452 . . . . . . . . . 10 (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
2119, 20mpbi 229 . . . . . . . . 9 Tr 𝒫 (TC‘𝑎)
22 trss 5276 . . . . . . . . 9 (Tr 𝒫 (TC‘𝑎) → (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎)))
2321, 22ax-mp 5 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎))
24 fvex 6904 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ∈ V
2524elpw 4606 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
26 sspwuni 5103 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
2723, 25, 263imtr3i 290 . . . . . . 7 (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
2818, 27eqsstrid 4030 . . . . . 6 (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎))
2928a1i 11 . . . . 5 (𝑐 ∈ ω → (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
306, 8, 10, 12, 17, 29finds 7888 . . . 4 (𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
31 vex 3478 . . . . . . . 8 𝑎 ∈ V
3213itunifn 10411 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
33 fndm 6652 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
3431, 32, 33mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
3534eleq2i 2825 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
36 ndmfv 6926 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
3735, 36sylnbir 330 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
38 0ss 4396 . . . . 5 ∅ ⊆ (TC‘𝑎)
3937, 38eqsstrdi 4036 . . . 4 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
4030, 39pm2.61i 182 . . 3 ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎)
414, 40vtoclg 3556 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
42 fv2prc 6936 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ∅)
43 0ss 4396 . . 3 ∅ ⊆ (TC‘𝐴)
4442, 43eqsstrdi 4036 . 2 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
4541, 44pm2.61i 182 1 ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wcel 2106  Vcvv 3474  wss 3948  c0 4322  𝒫 cpw 4602   cuni 4908  cmpt 5231  Tr wtr 5265  dom cdm 5676  cres 5678  suc csuc 6366   Fn wfn 6538  cfv 6543  ωcom 7854  reccrdg 8408  TCctc 9730
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724  ax-inf2 9635
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-tc 9731
This theorem is referenced by:  itunitc  10415
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