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Theorem itunitc1 10417
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 6890 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
21fveq1d 6892 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎)‘𝐵) = ((𝑈𝐴)‘𝐵))
3 fveq2 6890 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
42, 3sseq12d 4014 . . 3 (𝑎 = 𝐴 → (((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5 fveq2 6890 . . . . . 6 (𝑏 = ∅ → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘∅))
65sseq1d 4012 . . . . 5 (𝑏 = ∅ → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)))
7 fveq2 6890 . . . . . 6 (𝑏 = 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝑐))
87sseq1d 4012 . . . . 5 (𝑏 = 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9 fveq2 6890 . . . . . 6 (𝑏 = suc 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘suc 𝑐))
109sseq1d 4012 . . . . 5 (𝑏 = suc 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11 fveq2 6890 . . . . . 6 (𝑏 = 𝐵 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝐵))
1211sseq1d 4012 . . . . 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 10415 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
15 tcid 9736 . . . . . . 7 (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
1614, 15eqsstrd 4019 . . . . . 6 (𝑎 ∈ V → ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎))
1716elv 3478 . . . . 5 ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)
1813itunisuc 10416 . . . . . . 7 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
19 tctr 9737 . . . . . . . . . 10 Tr (TC‘𝑎)
20 pwtr 5451 . . . . . . . . . 10 (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
2119, 20mpbi 229 . . . . . . . . 9 Tr 𝒫 (TC‘𝑎)
22 trss 5275 . . . . . . . . 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 6903 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ∈ V
2524elpw 4605 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
26 sspwuni 5102 . . . . . . . 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 4029 . . . . . 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 7891 . . . 4 (𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
31 vex 3476 . . . . . . . 8 𝑎 ∈ V
3213itunifn 10414 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
33 fndm 6651 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
3431, 32, 33mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
3534eleq2i 2823 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
36 ndmfv 6925 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
3735, 36sylnbir 330 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
38 0ss 4395 . . . . 5 ∅ ⊆ (TC‘𝑎)
3937, 38eqsstrdi 4035 . . . 4 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
4030, 39pm2.61i 182 . . 3 ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎)
414, 40vtoclg 3541 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
42 fv2prc 6935 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ∅)
43 0ss 4395 . . 3 ∅ ⊆ (TC‘𝐴)
4442, 43eqsstrdi 4035 . 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 1539  wcel 2104  Vcvv 3472  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907  cmpt 5230  Tr wtr 5264  dom cdm 5675  cres 5677  suc csuc 6365   Fn wfn 6537  cfv 6542  ωcom 7857  reccrdg 8411  TCctc 9733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-tc 9734
This theorem is referenced by:  itunitc  10418
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