Theorem itunitc 10418

Description: The union of all union iterates creates the transitive closure; compare trcl 9725. (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 6891	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2		fveq2 6891	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
3	2	rneqd 5937	. . . . 5 ⊢ (𝑎 = 𝐴 → ran (𝑈‘𝑎) = ran (𝑈‘𝐴))
4	3	unieqd 4922	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑈‘𝑎) = ∪ ran (𝑈‘𝐴))
5	1, 4	eqeq12d 2748	. . 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 10415	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
8	7	elv 3480	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) = 𝑎
9		fvssunirn 6924	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) ⊆ ∪ ran (𝑈‘𝑎)
10	8, 9	eqsstrri 4017	. . . . 5 ⊢ 𝑎 ⊆ ∪ ran (𝑈‘𝑎)
11		dftr3 5271	. . . . . 6 ⊢ (Tr ∪ ran (𝑈‘𝑎) ↔ ∀𝑏 ∈ ∪ ran (𝑈‘𝑎)𝑏 ⊆ ∪ ran (𝑈‘𝑎))
12		vex 3478	. . . . . . . 8 ⊢ 𝑎 ∈ V
13	6	itunifn 10414	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
14		fnunirn 7255	. . . . . . . 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 4941	. . . . . . . . 9 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ((𝑈‘𝑎)‘𝑐))
17	6	itunisuc 10416	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
18		fvssunirn 6924	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) ⊆ ∪ ran (𝑈‘𝑎)
19	17, 18	eqsstrri 4017	. . . . . . . . 9 ⊢ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ ∪ ran (𝑈‘𝑎)
20	16, 19	sstrdi 3994	. . . . . . . 8 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
21	20	rexlimivw 3151	. . . . . . 7 ⊢ (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
22	15, 21	sylbi 216	. . . . . 6 ⊢ (𝑏 ∈ ∪ ran (𝑈‘𝑎) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
23	11, 22	mprgbir 3068	. . . . 5 ⊢ Tr ∪ ran (𝑈‘𝑎)
24		tcmin 9738	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)))
25	24	elv 3480	. . . . 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 4943	. . . . 5 ⊢ (∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈‘𝑎)𝑏 ⊆ (TC‘𝑎))
28		fvelrnb 6952	. . . . . . 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 10417	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑐 ∈ ω → ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
32		sseq1 4007	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) = 𝑏 → (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
33	31, 32	syl5ibcom 244	. . . . . . 7 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎)))
34	33	rexlimiv 3148	. . . . . 6 ⊢ (∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎))
35	29, 34	sylbi 216	. . . . 5 ⊢ (𝑏 ∈ ran (𝑈‘𝑎) → 𝑏 ⊆ (TC‘𝑎))
36	27, 35	mprgbir 3068	. . . 4 ⊢ ∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎)
37	26, 36	eqssi 3998	. . 3 ⊢ (TC‘𝑎) = ∪ ran (𝑈‘𝑎)
38	5, 37	vtoclg 3556	. 2 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
39		rn0 5925	. . . . 5 ⊢ ran ∅ = ∅
40	39	unieqi 4921	. . . 4 ⊢ ∪ ran ∅ = ∪ ∅
41		uni0 4939	. . . 4 ⊢ ∪ ∅ = ∅
42	40, 41	eqtr2i 2761	. . 3 ⊢ ∅ = ∪ ran ∅
43		fvprc 6883	. . 3 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
44		fvprc 6883	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
45	44	rneqd 5937	. . . 4 ⊢ (¬ 𝐴 ∈ V → ran (𝑈‘𝐴) = ran ∅)
46	45	unieqd 4922	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran (𝑈‘𝐴) = ∪ ran ∅)
47	42, 43, 46	3eqtr4a 2798	. 2 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
48	38, 47	pm2.61i 182	1 ⊢ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  ∪ cuni 4908   ↦ cmpt 5231  Tr wtr 5265  ran crn 5677   ↾ cres 5678  suc csuc 6366   Fn wfn 6538  ‘cfv 6543  ωcom 7857  reccrdg 8411  TCctc 9733

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-tc 9734

This theorem is referenced by:  hsmexlem5  10427