Theorem itunitc 10108

Description: The union of all union iterates creates the transitive closure; compare trcl 9417. (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 6756	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2		fveq2 6756	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
3	2	rneqd 5836	. . . . 5 ⊢ (𝑎 = 𝐴 → ran (𝑈‘𝑎) = ran (𝑈‘𝐴))
4	3	unieqd 4850	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑈‘𝑎) = ∪ ran (𝑈‘𝐴))
5	1, 4	eqeq12d 2754	. . 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 10105	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
8	7	elv 3428	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) = 𝑎
9		fvssunirn 6785	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) ⊆ ∪ ran (𝑈‘𝑎)
10	8, 9	eqsstrri 3952	. . . . 5 ⊢ 𝑎 ⊆ ∪ ran (𝑈‘𝑎)
11		dftr3 5191	. . . . . 6 ⊢ (Tr ∪ ran (𝑈‘𝑎) ↔ ∀𝑏 ∈ ∪ ran (𝑈‘𝑎)𝑏 ⊆ ∪ ran (𝑈‘𝑎))
12		vex 3426	. . . . . . . 8 ⊢ 𝑎 ∈ V
13	6	itunifn 10104	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
14		fnunirn 7108	. . . . . . . 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 4868	. . . . . . . . 9 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ((𝑈‘𝑎)‘𝑐))
17	6	itunisuc 10106	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
18		fvssunirn 6785	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) ⊆ ∪ ran (𝑈‘𝑎)
19	17, 18	eqsstrri 3952	. . . . . . . . 9 ⊢ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ ∪ ran (𝑈‘𝑎)
20	16, 19	sstrdi 3929	. . . . . . . 8 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
21	20	rexlimivw 3210	. . . . . . 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 3078	. . . . 5 ⊢ Tr ∪ ran (𝑈‘𝑎)
24		tcmin 9430	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)))
25	24	elv 3428	. . . . 5 ⊢ ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎))
26	10, 23, 25	mp2an 688	. . . 4 ⊢ (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)
27		unissb 4870	. . . . 5 ⊢ (∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈‘𝑎)𝑏 ⊆ (TC‘𝑎))
28		fvelrnb 6812	. . . . . . 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 10107	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑐 ∈ ω → ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
32		sseq1 3942	. . . . . . . 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 3208	. . . . . 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 3078	. . . 4 ⊢ ∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎)
37	26, 36	eqssi 3933	. . 3 ⊢ (TC‘𝑎) = ∪ ran (𝑈‘𝑎)
38	5, 37	vtoclg 3495	. 2 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
39		rn0 5824	. . . . 5 ⊢ ran ∅ = ∅
40	39	unieqi 4849	. . . 4 ⊢ ∪ ran ∅ = ∪ ∅
41		uni0 4866	. . . 4 ⊢ ∪ ∅ = ∅
42	40, 41	eqtr2i 2767	. . 3 ⊢ ∅ = ∪ ran ∅
43		fvprc 6748	. . 3 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
44		fvprc 6748	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
45	44	rneqd 5836	. . . 4 ⊢ (¬ 𝐴 ∈ V → ran (𝑈‘𝐴) = ran ∅)
46	45	unieqd 4850	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran (𝑈‘𝐴) = ∪ ran ∅)
47	42, 43, 46	3eqtr4a 2805	. 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 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836   ↦ cmpt 5153  Tr wtr 5187  ran crn 5581   ↾ cres 5582  suc csuc 6253   Fn wfn 6413  ‘cfv 6418  ωcom 7687  reccrdg 8211  TCctc 9425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-tc 9426

This theorem is referenced by:  hsmexlem5  10117