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Theorem itunitc 9435
Description: The union of all union iterates creates the transitive closure; compare trcl 8777. (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 6352 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2 fveq2 6352 . . . . . 6 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
32rneqd 5508 . . . . 5 (𝑎 = 𝐴 → ran (𝑈𝑎) = ran (𝑈𝐴))
43unieqd 4598 . . . 4 (𝑎 = 𝐴 ran (𝑈𝑎) = ran (𝑈𝐴))
51, 4eqeq12d 2775 . . 3 (𝑎 = 𝐴 → ((TC‘𝑎) = ran (𝑈𝑎) ↔ (TC‘𝐴) = ran (𝑈𝐴)))
6 vex 3343 . . . . . . 7 𝑎 ∈ V
7 ituni.u . . . . . . . 8 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
87ituni0 9432 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
96, 8ax-mp 5 . . . . . 6 ((𝑈𝑎)‘∅) = 𝑎
10 fvssunirn 6378 . . . . . 6 ((𝑈𝑎)‘∅) ⊆ ran (𝑈𝑎)
119, 10eqsstr3i 3777 . . . . 5 𝑎 ran (𝑈𝑎)
12 dftr3 4908 . . . . . 6 (Tr ran (𝑈𝑎) ↔ ∀𝑏 ran (𝑈𝑎)𝑏 ran (𝑈𝑎))
137itunifn 9431 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
14 fnunirn 6674 . . . . . . . 8 ((𝑈𝑎) Fn ω → (𝑏 ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐)))
156, 13, 14mp2b 10 . . . . . . 7 (𝑏 ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐))
16 elssuni 4619 . . . . . . . . 9 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ((𝑈𝑎)‘𝑐))
177itunisuc 9433 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
18 fvssunirn 6378 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) ⊆ ran (𝑈𝑎)
1917, 18eqsstr3i 3777 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ ran (𝑈𝑎)
2016, 19syl6ss 3756 . . . . . . . 8 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2120rexlimivw 3167 . . . . . . 7 (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2215, 21sylbi 207 . . . . . 6 (𝑏 ran (𝑈𝑎) → 𝑏 ran (𝑈𝑎))
2312, 22mprgbir 3065 . . . . 5 Tr ran (𝑈𝑎)
24 tcmin 8790 . . . . . 6 (𝑎 ∈ V → ((𝑎 ran (𝑈𝑎) ∧ Tr ran (𝑈𝑎)) → (TC‘𝑎) ⊆ ran (𝑈𝑎)))
256, 24ax-mp 5 . . . . 5 ((𝑎 ran (𝑈𝑎) ∧ Tr ran (𝑈𝑎)) → (TC‘𝑎) ⊆ ran (𝑈𝑎))
2611, 23, 25mp2an 710 . . . 4 (TC‘𝑎) ⊆ ran (𝑈𝑎)
27 unissb 4621 . . . . 5 ( ran (𝑈𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈𝑎)𝑏 ⊆ (TC‘𝑎))
28 fvelrnb 6405 . . . . . . 7 ((𝑈𝑎) Fn ω → (𝑏 ∈ ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏))
296, 13, 28mp2b 10 . . . . . 6 (𝑏 ∈ ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏)
307itunitc1 9434 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)
3130a1i 11 . . . . . . . 8 (𝑐 ∈ ω → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
32 sseq1 3767 . . . . . . . 8 (((𝑈𝑎)‘𝑐) = 𝑏 → (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
3331, 32syl5ibcom 235 . . . . . . 7 (𝑐 ∈ ω → (((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎)))
3433rexlimiv 3165 . . . . . 6 (∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎))
3529, 34sylbi 207 . . . . 5 (𝑏 ∈ ran (𝑈𝑎) → 𝑏 ⊆ (TC‘𝑎))
3627, 35mprgbir 3065 . . . 4 ran (𝑈𝑎) ⊆ (TC‘𝑎)
3726, 36eqssi 3760 . . 3 (TC‘𝑎) = ran (𝑈𝑎)
385, 37vtoclg 3406 . 2 (𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
39 rn0 5532 . . . . 5 ran ∅ = ∅
4039unieqi 4597 . . . 4 ran ∅ =
41 uni0 4617 . . . 4 ∅ = ∅
4240, 41eqtr2i 2783 . . 3 ∅ = ran ∅
43 fvprc 6346 . . 3 𝐴 ∈ V → (TC‘𝐴) = ∅)
44 fvprc 6346 . . . . 5 𝐴 ∈ V → (𝑈𝐴) = ∅)
4544rneqd 5508 . . . 4 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4645unieqd 4598 . . 3 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4742, 43, 463eqtr4a 2820 . 2 𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
4838, 47pm2.61i 176 1 (TC‘𝐴) = ran (𝑈𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  wss 3715  c0 4058   cuni 4588  cmpt 4881  Tr wtr 4904  ran crn 5267  cres 5268  suc csuc 5886   Fn wfn 6044  cfv 6049  ωcom 7230  reccrdg 7674  TCctc 8785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-tc 8786
This theorem is referenced by:  hsmexlem5  9444
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