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Theorem itunitc 10461
Description: The union of all union iterates creates the transitive closure; compare trcl 9768. (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
itunitc (TC‘𝐴) = ran (𝑈𝐴)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunitc
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2 fveq2 6906 . . . . . 6 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
32rneqd 5949 . . . . 5 (𝑎 = 𝐴 → ran (𝑈𝑎) = ran (𝑈𝐴))
43unieqd 4920 . . . 4 (𝑎 = 𝐴 ran (𝑈𝑎) = ran (𝑈𝐴))
51, 4eqeq12d 2753 . . 3 (𝑎 = 𝐴 → ((TC‘𝑎) = ran (𝑈𝑎) ↔ (TC‘𝐴) = ran (𝑈𝐴)))
6 ituni.u . . . . . . . 8 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
76ituni0 10458 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
87elv 3485 . . . . . 6 ((𝑈𝑎)‘∅) = 𝑎
9 fvssunirn 6939 . . . . . 6 ((𝑈𝑎)‘∅) ⊆ ran (𝑈𝑎)
108, 9eqsstrri 4031 . . . . 5 𝑎 ran (𝑈𝑎)
11 dftr3 5265 . . . . . 6 (Tr ran (𝑈𝑎) ↔ ∀𝑏 ran (𝑈𝑎)𝑏 ran (𝑈𝑎))
12 vex 3484 . . . . . . . 8 𝑎 ∈ V
136itunifn 10457 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
14 fnunirn 7274 . . . . . . . 8 ((𝑈𝑎) Fn ω → (𝑏 ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐)))
1512, 13, 14mp2b 10 . . . . . . 7 (𝑏 ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐))
16 elssuni 4937 . . . . . . . . 9 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ((𝑈𝑎)‘𝑐))
176itunisuc 10459 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
18 fvssunirn 6939 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) ⊆ ran (𝑈𝑎)
1917, 18eqsstrri 4031 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ ran (𝑈𝑎)
2016, 19sstrdi 3996 . . . . . . . 8 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2120rexlimivw 3151 . . . . . . 7 (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2215, 21sylbi 217 . . . . . 6 (𝑏 ran (𝑈𝑎) → 𝑏 ran (𝑈𝑎))
2311, 22mprgbir 3068 . . . . 5 Tr ran (𝑈𝑎)
24 tcmin 9781 . . . . . 6 (𝑎 ∈ V → ((𝑎 ran (𝑈𝑎) ∧ Tr ran (𝑈𝑎)) → (TC‘𝑎) ⊆ ran (𝑈𝑎)))
2524elv 3485 . . . . 5 ((𝑎 ran (𝑈𝑎) ∧ Tr ran (𝑈𝑎)) → (TC‘𝑎) ⊆ ran (𝑈𝑎))
2610, 23, 25mp2an 692 . . . 4 (TC‘𝑎) ⊆ ran (𝑈𝑎)
27 unissb 4939 . . . . 5 ( ran (𝑈𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈𝑎)𝑏 ⊆ (TC‘𝑎))
28 fvelrnb 6969 . . . . . . 7 ((𝑈𝑎) Fn ω → (𝑏 ∈ ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏))
2912, 13, 28mp2b 10 . . . . . 6 (𝑏 ∈ ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏)
306itunitc1 10460 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)
3130a1i 11 . . . . . . . 8 (𝑐 ∈ ω → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
32 sseq1 4009 . . . . . . . 8 (((𝑈𝑎)‘𝑐) = 𝑏 → (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
3331, 32syl5ibcom 245 . . . . . . 7 (𝑐 ∈ ω → (((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎)))
3433rexlimiv 3148 . . . . . 6 (∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎))
3529, 34sylbi 217 . . . . 5 (𝑏 ∈ ran (𝑈𝑎) → 𝑏 ⊆ (TC‘𝑎))
3627, 35mprgbir 3068 . . . 4 ran (𝑈𝑎) ⊆ (TC‘𝑎)
3726, 36eqssi 4000 . . 3 (TC‘𝑎) = ran (𝑈𝑎)
385, 37vtoclg 3554 . 2 (𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
39 rn0 5936 . . . . 5 ran ∅ = ∅
4039unieqi 4919 . . . 4 ran ∅ =
41 uni0 4935 . . . 4 ∅ = ∅
4240, 41eqtr2i 2766 . . 3 ∅ = ran ∅
43 fvprc 6898 . . 3 𝐴 ∈ V → (TC‘𝐴) = ∅)
44 fvprc 6898 . . . . 5 𝐴 ∈ V → (𝑈𝐴) = ∅)
4544rneqd 5949 . . . 4 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4645unieqd 4920 . . 3 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4742, 43, 463eqtr4a 2803 . 2 𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
4838, 47pm2.61i 182 1 (TC‘𝐴) = ran (𝑈𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  c0 4333   cuni 4907  cmpt 5225  Tr wtr 5259  ran crn 5686  cres 5687  suc csuc 6386   Fn wfn 6556  cfv 6561  ωcom 7887  reccrdg 8449  TCctc 9776
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-tc 9777
This theorem is referenced by:  hsmexlem5  10470
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