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Theorem itunitc1 9885
 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 6662 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
21fveq1d 6664 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎)‘𝐵) = ((𝑈𝐴)‘𝐵))
3 fveq2 6662 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
42, 3sseq12d 3927 . . 3 (𝑎 = 𝐴 → (((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎) ↔ ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)))
5 fveq2 6662 . . . . . 6 (𝑏 = ∅ → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘∅))
65sseq1d 3925 . . . . 5 (𝑏 = ∅ → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)))
7 fveq2 6662 . . . . . 6 (𝑏 = 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝑐))
87sseq1d 3925 . . . . 5 (𝑏 = 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)))
9 fveq2 6662 . . . . . 6 (𝑏 = suc 𝑐 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘suc 𝑐))
109sseq1d 3925 . . . . 5 (𝑏 = suc 𝑐 → (((𝑈𝑎)‘𝑏) ⊆ (TC‘𝑎) ↔ ((𝑈𝑎)‘suc 𝑐) ⊆ (TC‘𝑎)))
11 fveq2 6662 . . . . . 6 (𝑏 = 𝐵 → ((𝑈𝑎)‘𝑏) = ((𝑈𝑎)‘𝐵))
1211sseq1d 3925 . . . . 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 9883 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
15 tcid 9219 . . . . . . 7 (𝑎 ∈ V → 𝑎 ⊆ (TC‘𝑎))
1614, 15eqsstrd 3932 . . . . . 6 (𝑎 ∈ V → ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎))
1716elv 3415 . . . . 5 ((𝑈𝑎)‘∅) ⊆ (TC‘𝑎)
1813itunisuc 9884 . . . . . . 7 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
19 tctr 9220 . . . . . . . . . 10 Tr (TC‘𝑎)
20 pwtr 5316 . . . . . . . . . 10 (Tr (TC‘𝑎) ↔ Tr 𝒫 (TC‘𝑎))
2119, 20mpbi 233 . . . . . . . . 9 Tr 𝒫 (TC‘𝑎)
22 trss 5150 . . . . . . . . 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 6675 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ∈ V
2524elpw 4501 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ∈ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
26 sspwuni 4990 . . . . . . . 8 (((𝑈𝑎)‘𝑐) ⊆ 𝒫 (TC‘𝑎) ↔ ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
2723, 25, 263imtr3i 294 . . . . . . 7 (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
2818, 27eqsstrid 3942 . . . . . 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 7613 . . . 4 (𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
31 vex 3413 . . . . . . . 8 𝑎 ∈ V
3213itunifn 9882 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
33 fndm 6440 . . . . . . . 8 ((𝑈𝑎) Fn ω → dom (𝑈𝑎) = ω)
3431, 32, 33mp2b 10 . . . . . . 7 dom (𝑈𝑎) = ω
3534eleq2i 2843 . . . . . 6 (𝐵 ∈ dom (𝑈𝑎) ↔ 𝐵 ∈ ω)
36 ndmfv 6692 . . . . . 6 𝐵 ∈ dom (𝑈𝑎) → ((𝑈𝑎)‘𝐵) = ∅)
3735, 36sylnbir 334 . . . . 5 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) = ∅)
38 0ss 4295 . . . . 5 ∅ ⊆ (TC‘𝑎)
3937, 38eqsstrdi 3948 . . . 4 𝐵 ∈ ω → ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎))
4030, 39pm2.61i 185 . . 3 ((𝑈𝑎)‘𝐵) ⊆ (TC‘𝑎)
414, 40vtoclg 3487 . 2 (𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
42 fv2prc 6702 . . 3 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) = ∅)
43 0ss 4295 . . 3 ∅ ⊆ (TC‘𝐴)
4442, 43eqsstrdi 3948 . 2 𝐴 ∈ V → ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴))
4541, 44pm2.61i 185 1 ((𝑈𝐴)‘𝐵) ⊆ (TC‘𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  ∪ cuni 4801   ↦ cmpt 5115  Tr wtr 5141  dom cdm 5527   ↾ cres 5529  suc csuc 6175   Fn wfn 6334  ‘cfv 6339  ωcom 7584  reccrdg 8060  TCctc 9216 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464  ax-inf2 9142 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7585  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-tc 9217 This theorem is referenced by:  itunitc  9886
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