Theorem itunitc 9126

Description: The union of all union iterates creates the transitive closure; compare trcl 8487. (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 6103	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2		fveq2 6103	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
3	2	rneqd 5274	. . . . 5 ⊢ (𝑎 = 𝐴 → ran (𝑈‘𝑎) = ran (𝑈‘𝐴))
4	3	unieqd 4382	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑈‘𝑎) = ∪ ran (𝑈‘𝐴))
5	1, 4	eqeq12d 2625	. . 3 ⊢ (𝑎 = 𝐴 → ((TC‘𝑎) = ∪ ran (𝑈‘𝑎) ↔ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)))
6		vex 3176	. . . . . . 7 ⊢ 𝑎 ∈ V
7		ituni.u	. . . . . . . 8 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
8	7	ituni0 9123	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
9	6, 8	ax-mp 5	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) = 𝑎
10		fvssunirn 6127	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) ⊆ ∪ ran (𝑈‘𝑎)
11	9, 10	eqsstr3i 3599	. . . . 5 ⊢ 𝑎 ⊆ ∪ ran (𝑈‘𝑎)
12		dftr3 4684	. . . . . 6 ⊢ (Tr ∪ ran (𝑈‘𝑎) ↔ ∀𝑏 ∈ ∪ ran (𝑈‘𝑎)𝑏 ⊆ ∪ ran (𝑈‘𝑎))
13	7	itunifn 9122	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
14		fnunirn 6415	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → (𝑏 ∈ ∪ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐)))
15	6, 13, 14	mp2b 10	. . . . . . 7 ⊢ (𝑏 ∈ ∪ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐))
16		elssuni 4403	. . . . . . . . 9 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ((𝑈‘𝑎)‘𝑐))
17	7	itunisuc 9124	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
18		fvssunirn 6127	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) ⊆ ∪ ran (𝑈‘𝑎)
19	17, 18	eqsstr3i 3599	. . . . . . . . 9 ⊢ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ ∪ ran (𝑈‘𝑎)
20	16, 19	syl6ss 3580	. . . . . . . 8 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
21	20	rexlimivw 3011	. . . . . . 7 ⊢ (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
22	15, 21	sylbi 206	. . . . . 6 ⊢ (𝑏 ∈ ∪ ran (𝑈‘𝑎) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
23	12, 22	mprgbir 2911	. . . . 5 ⊢ Tr ∪ ran (𝑈‘𝑎)
24		tcmin 8500	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)))
25	6, 24	ax-mp 5	. . . . 5 ⊢ ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎))
26	11, 23, 25	mp2an 704	. . . 4 ⊢ (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)
27		unissb 4405	. . . . 5 ⊢ (∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈‘𝑎)𝑏 ⊆ (TC‘𝑎))
28		fvelrnb 6153	. . . . . . 7 ⊢ ((𝑈‘𝑎) Fn ω → (𝑏 ∈ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏))
29	6, 13, 28	mp2b 10	. . . . . 6 ⊢ (𝑏 ∈ ran (𝑈‘𝑎) ↔ ∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏)
30	7	itunitc1 9125	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑐 ∈ ω → ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
32		sseq1 3589	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) = 𝑏 → (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
33	31, 32	syl5ibcom 234	. . . . . . 7 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎)))
34	33	rexlimiv 3009	. . . . . 6 ⊢ (∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎))
35	29, 34	sylbi 206	. . . . 5 ⊢ (𝑏 ∈ ran (𝑈‘𝑎) → 𝑏 ⊆ (TC‘𝑎))
36	27, 35	mprgbir 2911	. . . 4 ⊢ ∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎)
37	26, 36	eqssi 3584	. . 3 ⊢ (TC‘𝑎) = ∪ ran (𝑈‘𝑎)
38	5, 37	vtoclg 3239	. 2 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
39		rn0 5298	. . . . 5 ⊢ ran ∅ = ∅
40	39	unieqi 4381	. . . 4 ⊢ ∪ ran ∅ = ∪ ∅
41		uni0 4401	. . . 4 ⊢ ∪ ∅ = ∅
42	40, 41	eqtr2i 2633	. . 3 ⊢ ∅ = ∪ ran ∅
43		fvprc 6097	. . 3 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
44		fvprc 6097	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
45	44	rneqd 5274	. . . 4 ⊢ (¬ 𝐴 ∈ V → ran (𝑈‘𝐴) = ran ∅)
46	45	unieqd 4382	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran (𝑈‘𝐴) = ∪ ran ∅)
47	42, 43, 46	3eqtr4a 2670	. 2 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
48	38, 47	pm2.61i 175	1 ⊢ (TC‘𝐴) = ∪ ran (𝑈‘𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   ↦ cmpt 4643  Tr wtr 4680  ran crn 5039   ↾ cres 5040  suc csuc 5642   Fn wfn 5799  ‘cfv 5804  ωcom 6957  reccrdg 7392  TCctc 8495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-tc 8496

This theorem is referenced by:  hsmexlem5  9135