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Theorem itunitc1 10411
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 6888 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
21fveq1d 6890 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎)‘𝐵) = ((𝑈𝐴)‘𝐵))
3 fveq2 6888 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
42, 3sseq12d 4014 . . 3 (𝑎 = 𝐴 → (((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5 fveq2 6888 . . . . . 6 (𝑏 = ∅ → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘∅))
65sseq1d 4012 . . . . 5 (𝑏 = ∅ → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)))
7 fveq2 6888 . . . . . 6 (𝑏 = 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝑐))
87sseq1d 4012 . . . . 5 (𝑏 = 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9 fveq2 6888 . . . . . 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 6888 . . . . . 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 10409 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
15 tcid 9730 . . . . . . 7 (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
1614, 15eqsstrd 4019 . . . . . 6 (𝑎 ∈ V → ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎))
1716elv 3481 . . . . 5 ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)
1813itunisuc 10410 . . . . . . 7 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
19 tctr 9731 . . . . . . . . . 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 6901 . . . . . . . . 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 291 . . . . . . 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 7884 . . . 4 (𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
31 vex 3479 . . . . . . . 8 𝑎 ∈ V
3213itunifn 10408 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
33 fndm 6649 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
3431, 32, 33mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
3534eleq2i 2826 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
36 ndmfv 6923 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
3735, 36sylnbir 331 . . . . 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 3556 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
42 fv2prc 6933 . . 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 1542  wcel 2107  Vcvv 3475  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907  cmpt 5230  Tr wtr 5264  dom cdm 5675  cres 5677  suc csuc 6363   Fn wfn 6535  cfv 6540  ωcom 7850  reccrdg 8404  TCctc 9727
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-tc 9728
This theorem is referenced by:  itunitc  10412
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