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Theorem itunitc1 9227
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 6178 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
21fveq1d 6180 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎)‘𝐵) = ((𝑈𝐴)‘𝐵))
3 fveq2 6178 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
42, 3sseq12d 3626 . . 3 (𝑎 = 𝐴 → (((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5 fveq2 6178 . . . . . 6 (𝑏 = ∅ → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘∅))
65sseq1d 3624 . . . . 5 (𝑏 = ∅ → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)))
7 fveq2 6178 . . . . . 6 (𝑏 = 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝑐))
87sseq1d 3624 . . . . 5 (𝑏 = 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9 fveq2 6178 . . . . . 6 (𝑏 = suc 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘suc 𝑐))
109sseq1d 3624 . . . . 5 (𝑏 = suc 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11 fveq2 6178 . . . . . 6 (𝑏 = 𝐵 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝐵))
1211sseq1d 3624 . . . . 5 (𝑏 = 𝐵 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎)))
13 vex 3198 . . . . . 6 𝑎 ∈ V
14 ituni.u . . . . . . . 8 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
1514ituni0 9225 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
16 tcid 8600 . . . . . . 7 (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
1715, 16eqsstrd 3631 . . . . . 6 (𝑎 ∈ V → ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎))
1813, 17ax-mp 5 . . . . 5 ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)
1914itunisuc 9226 . . . . . . 7 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
20 tctr 8601 . . . . . . . . . 10 Tr (TC‘𝑎)
21 pwtr 4912 . . . . . . . . . 10 (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
2220, 21mpbi 220 . . . . . . . . 9 Tr 𝒫 (TC‘𝑎)
23 trss 4752 . . . . . . . . 9 (Tr 𝒫 (TC‘𝑎) → (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎)))
2422, 23ax-mp 5 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎))
25 fvex 6188 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ∈ V
2625elpw 4155 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
27 sspwuni 4602 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
2824, 26, 273imtr3i 280 . . . . . . 7 (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
2919, 28syl5eqss 3641 . . . . . 6 (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎))
3029a1i 11 . . . . 5 (𝑐 ∈ ω → (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
316, 8, 10, 12, 18, 30finds 7077 . . . 4 (𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
3214itunifn 9224 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
33 fndm 5978 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
3413, 32, 33mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
3534eleq2i 2691 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
36 ndmfv 6205 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
3735, 36sylnbir 321 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
38 0ss 3963 . . . . 5 ∅ ⊆ (TC‘𝑎)
3937, 38syl6eqss 3647 . . . 4 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
4031, 39pm2.61i 176 . . 3 ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎)
414, 40vtoclg 3261 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
42 fvprc 6172 . . . . 5 𝐴 ∈ V → (𝑈𝐴) = ∅)
4342fveq1d 6180 . . . 4 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = (∅‘𝐵))
44 0fv 6214 . . . 4 (∅‘𝐵) = ∅
4543, 44syl6eq 2670 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ∅)
46 0ss 3963 . . 3 ∅ ⊆ (TC‘𝐴)
4745, 46syl6eqss 3647 . 2 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
4841, 47pm2.61i 176 1 ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  wss 3567  c0 3907  𝒫 cpw 4149   cuni 4427  cmpt 4720  Tr wtr 4743  dom cdm 5104  cres 5106  suc csuc 5713   Fn wfn 5871  cfv 5876  ωcom 7050  reccrdg 7490  TCctc 8597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-tc 8598
This theorem is referenced by:  itunitc  9228
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