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Theorem itunitc 9187
Description: The union of all union iterates creates the transitive closure; compare trcl 8548. (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 6148 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2 fveq2 6148 . . . . . 6 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
32rneqd 5313 . . . . 5 (𝑎 = 𝐴 → ran (𝑈𝑎) = ran (𝑈𝐴))
43unieqd 4412 . . . 4 (𝑎 = 𝐴 ran (𝑈𝑎) = ran (𝑈𝐴))
51, 4eqeq12d 2636 . . 3 (𝑎 = 𝐴 → ((TC‘𝑎) = ran (𝑈𝑎) ↔ (TC‘𝐴) = ran (𝑈𝐴)))
6 vex 3189 . . . . . . 7 𝑎 ∈ V
7 ituni.u . . . . . . . 8 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
87ituni0 9184 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
96, 8ax-mp 5 . . . . . 6 ((𝑈𝑎)‘∅) = 𝑎
10 fvssunirn 6174 . . . . . 6 ((𝑈𝑎)‘∅) ⊆ ran (𝑈𝑎)
119, 10eqsstr3i 3615 . . . . 5 𝑎 ran (𝑈𝑎)
12 dftr3 4716 . . . . . 6 (Tr ran (𝑈𝑎) ↔ ∀𝑏 ran (𝑈𝑎)𝑏 ran (𝑈𝑎))
137itunifn 9183 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
14 fnunirn 6465 . . . . . . . 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 4433 . . . . . . . . 9 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ((𝑈𝑎)‘𝑐))
177itunisuc 9185 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
18 fvssunirn 6174 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) ⊆ ran (𝑈𝑎)
1917, 18eqsstr3i 3615 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ ran (𝑈𝑎)
2016, 19syl6ss 3595 . . . . . . . 8 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2120rexlimivw 3022 . . . . . . 7 (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2215, 21sylbi 207 . . . . . 6 (𝑏 ran (𝑈𝑎) → 𝑏 ran (𝑈𝑎))
2312, 22mprgbir 2922 . . . . 5 Tr ran (𝑈𝑎)
24 tcmin 8561 . . . . . 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 707 . . . 4 (TC‘𝑎) ⊆ ran (𝑈𝑎)
27 unissb 4435 . . . . 5 ( ran (𝑈𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈𝑎)𝑏 ⊆ (TC‘𝑎))
28 fvelrnb 6200 . . . . . . 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 9186 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)
3130a1i 11 . . . . . . . 8 (𝑐 ∈ ω → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
32 sseq1 3605 . . . . . . . 8 (((𝑈𝑎)‘𝑐) = 𝑏 → (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
3331, 32syl5ibcom 235 . . . . . . 7 (𝑐 ∈ ω → (((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎)))
3433rexlimiv 3020 . . . . . 6 (∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎))
3529, 34sylbi 207 . . . . 5 (𝑏 ∈ ran (𝑈𝑎) → 𝑏 ⊆ (TC‘𝑎))
3627, 35mprgbir 2922 . . . 4 ran (𝑈𝑎) ⊆ (TC‘𝑎)
3726, 36eqssi 3599 . . 3 (TC‘𝑎) = ran (𝑈𝑎)
385, 37vtoclg 3252 . 2 (𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
39 rn0 5337 . . . . 5 ran ∅ = ∅
4039unieqi 4411 . . . 4 ran ∅ =
41 uni0 4431 . . . 4 ∅ = ∅
4240, 41eqtr2i 2644 . . 3 ∅ = ran ∅
43 fvprc 6142 . . 3 𝐴 ∈ V → (TC‘𝐴) = ∅)
44 fvprc 6142 . . . . 5 𝐴 ∈ V → (𝑈𝐴) = ∅)
4544rneqd 5313 . . . 4 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4645unieqd 4412 . . 3 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4742, 43, 463eqtr4a 2681 . 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 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wss 3555  c0 3891   cuni 4402  cmpt 4673  Tr wtr 4712  ran crn 5075  cres 5076  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  reccrdg 7450  TCctc 8556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-tc 8557
This theorem is referenced by:  hsmexlem5  9196
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