Theorem itunitc 9845

Description: The union of all union iterates creates the transitive closure; compare trcl 9172. (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 (𝑈‘𝐴)

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.

Step	Hyp	Ref	Expression
1		fveq2 6672	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2		fveq2 6672	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
3	2	rneqd 5810	. . . . 5 ⊢ (𝑎 = 𝐴 → ran (𝑈‘𝑎) = ran (𝑈‘𝐴))
4	3	unieqd 4854	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑈‘𝑎) = ∪ ran (𝑈‘𝐴))
5	1, 4	eqeq12d 2839	. . 3 ⊢ (𝑎 = 𝐴 → ((TC‘𝑎) = ∪ ran (𝑈‘𝑎) ↔ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)))
6		ituni.u	. . . . . . . 8 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
7	6	ituni0 9842	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
8	7	elv 3501	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) = 𝑎
9		fvssunirn 6701	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) ⊆ ∪ ran (𝑈‘𝑎)
10	8, 9	eqsstrri 4004	. . . . 5 ⊢ 𝑎 ⊆ ∪ ran (𝑈‘𝑎)
11		dftr3 5178	. . . . . 6 ⊢ (Tr ∪ ran (𝑈‘𝑎) ↔ ∀𝑏 ∈ ∪ ran (𝑈‘𝑎)𝑏 ⊆ ∪ ran (𝑈‘𝑎))
12		vex 3499	. . . . . . . 8 ⊢ 𝑎 ∈ V
13	6	itunifn 9841	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
14		fnunirn 7014	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → (𝑏 ∈ ∪ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐)))
15	12, 13, 14	mp2b 10	. . . . . . 7 ⊢ (𝑏 ∈ ∪ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐))
16		elssuni 4870	. . . . . . . . 9 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ((𝑈‘𝑎)‘𝑐))
17	6	itunisuc 9843	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
18		fvssunirn 6701	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) ⊆ ∪ ran (𝑈‘𝑎)
19	17, 18	eqsstrri 4004	. . . . . . . . 9 ⊢ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ ∪ ran (𝑈‘𝑎)
20	16, 19	sstrdi 3981	. . . . . . . 8 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
21	20	rexlimivw 3284	. . . . . . 7 ⊢ (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
22	15, 21	sylbi 219	. . . . . 6 ⊢ (𝑏 ∈ ∪ ran (𝑈‘𝑎) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
23	11, 22	mprgbir 3155	. . . . 5 ⊢ Tr ∪ ran (𝑈‘𝑎)
24		tcmin 9185	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)))
25	24	elv 3501	. . . . 5 ⊢ ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎))
26	10, 23, 25	mp2an 690	. . . 4 ⊢ (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)
27		unissb 4872	. . . . 5 ⊢ (∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈‘𝑎)𝑏 ⊆ (TC‘𝑎))
28		fvelrnb 6728	. . . . . . 7 ⊢ ((𝑈‘𝑎) Fn ω → (𝑏 ∈ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏))
29	12, 13, 28	mp2b 10	. . . . . 6 ⊢ (𝑏 ∈ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏)
30	6	itunitc1 9844	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑐 ∈ ω → ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
32		sseq1 3994	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) = 𝑏 → (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
33	31, 32	syl5ibcom 247	. . . . . . 7 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎)))
34	33	rexlimiv 3282	. . . . . 6 ⊢ (∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎))
35	29, 34	sylbi 219	. . . . 5 ⊢ (𝑏 ∈ ran (𝑈‘𝑎) → 𝑏 ⊆ (TC‘𝑎))
36	27, 35	mprgbir 3155	. . . 4 ⊢ ∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎)
37	26, 36	eqssi 3985	. . 3 ⊢ (TC‘𝑎) = ∪ ran (𝑈‘𝑎)
38	5, 37	vtoclg 3569	. 2 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
39		rn0 5798	. . . . 5 ⊢ ran ∅ = ∅
40	39	unieqi 4853	. . . 4 ⊢ ∪ ran ∅ = ∪ ∅
41		uni0 4868	. . . 4 ⊢ ∪ ∅ = ∅
42	40, 41	eqtr2i 2847	. . 3 ⊢ ∅ = ∪ ran ∅
43		fvprc 6665	. . 3 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
44		fvprc 6665	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
45	44	rneqd 5810	. . . 4 ⊢ (¬ 𝐴 ∈ V → ran (𝑈‘𝐴) = ran ∅)
46	45	unieqd 4854	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran (𝑈‘𝐴) = ∪ ran ∅)
47	42, 43, 46	3eqtr4a 2884	. 2 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
48	38, 47	pm2.61i 184	1 ⊢ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  ∪ cuni 4840   ↦ cmpt 5148  Tr wtr 5174  ran crn 5558   ↾ cres 5559  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  ωcom 7582  reccrdg 8047  TCctc 9180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-tc 9181

This theorem is referenced by:  hsmexlem5  9854