Theorem itunitc 10404

Description: The union of all union iterates creates the transitive closure; compare trcl 9696. (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 6881	. . . 4 ⊢ (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2		fveq2 6881	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑈‘𝑎) = (𝑈‘𝐴))
3	2	rneqd 5928	. . . . 5 ⊢ (𝑎 = 𝐴 → ran (𝑈‘𝑎) = ran (𝑈‘𝐴))
4	3	unieqd 4884	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ ran (𝑈‘𝑎) = ∪ ran (𝑈‘𝐴))
5	1, 4	eqeq12d 2777	. . 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 10401	. . . . . . 7 ⊢ (𝑎 ∈ V → ((𝑈‘𝑎)‘∅) = 𝑎)
8	7	elv 3458	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) = 𝑎
9		fvssunirn 6912	. . . . . 6 ⊢ ((𝑈‘𝑎)‘∅) ⊆ ∪ ran (𝑈‘𝑎)
10	8, 9	eqsstrri 3983	. . . . 5 ⊢ 𝑎 ⊆ ∪ ran (𝑈‘𝑎)
11		dftr3 5222	. . . . . 6 ⊢ (Tr ∪ ran (𝑈‘𝑎) ↔ ∀𝑏 ∈ ∪ ran (𝑈‘𝑎)𝑏 ⊆ ∪ ran (𝑈‘𝑎))
12		vex 3457	. . . . . . . 8 ⊢ 𝑎 ∈ V
13	6	itunifn 10400	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
14		fnunirn 7251	. . . . . . . 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 4903	. . . . . . . . 9 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ((𝑈‘𝑎)‘𝑐))
17	6	itunisuc 10402	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
18		fvssunirn 6912	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) ⊆ ∪ ran (𝑈‘𝑎)
19	17, 18	eqsstrri 3983	. . . . . . . . 9 ⊢ ∪ ((𝑈‘𝑎)‘𝑐) ⊆ ∪ ran (𝑈‘𝑎)
20	16, 19	sstrdi 3948	. . . . . . . 8 ⊢ (𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
21	20	rexlimivw 3160	. . . . . . 7 ⊢ (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈‘𝑎)‘𝑐) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
22	15, 21	sylbi 220	. . . . . 6 ⊢ (𝑏 ∈ ∪ ran (𝑈‘𝑎) → 𝑏 ⊆ ∪ ran (𝑈‘𝑎))
23	11, 22	mprgbir 3084	. . . . 5 ⊢ Tr ∪ ran (𝑈‘𝑎)
24		tcmin 9707	. . . . . 6 ⊢ (𝑎 ∈ V → ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)))
25	24	elv 3458	. . . . 5 ⊢ ((𝑎 ⊆ ∪ ran (𝑈‘𝑎) ∧ Tr ∪ ran (𝑈‘𝑎)) → (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎))
26	10, 23, 25	mp2an 704	. . . 4 ⊢ (TC‘𝑎) ⊆ ∪ ran (𝑈‘𝑎)
27		unissb 4905	. . . . 5 ⊢ (∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈‘𝑎)𝑏 ⊆ (TC‘𝑎))
28		fvelrnb 6941	. . . . . . 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 10403	. . . . . . . . 9 ⊢ ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎)
31	30	a1i 11	. . . . . . . 8 ⊢ (𝑐 ∈ ω → ((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎))
32		sseq1 3961	. . . . . . . 8 ⊢ (((𝑈‘𝑎)‘𝑐) = 𝑏 → (((𝑈‘𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
33	31, 32	syl5ibcom 248	. . . . . . 7 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎)))
34	33	rexlimiv 3157	. . . . . 6 ⊢ (∃𝑐 ∈ ω ((𝑈‘𝑎)‘𝑐) = 𝑏 → 𝑏 ⊆ (TC‘𝑎))
35	29, 34	sylbi 220	. . . . 5 ⊢ (𝑏 ∈ ran (𝑈‘𝑎) → 𝑏 ⊆ (TC‘𝑎))
36	27, 35	mprgbir 3084	. . . 4 ⊢ ∪ ran (𝑈‘𝑎) ⊆ (TC‘𝑎)
37	26, 36	eqssi 3952	. . 3 ⊢ (TC‘𝑎) = ∪ ran (𝑈‘𝑎)
38	5, 37	vtoclg 3521	. 2 ⊢ (𝐴 ∈ V → (TC‘𝐴) = ∪ ran (𝑈‘𝐴))
39		rn0 5916	. . . . 5 ⊢ ran ∅ = ∅
40	39	unieqi 4883	. . . 4 ⊢ ∪ ran ∅ = ∪ ∅
41		uni0 4900	. . . 4 ⊢ ∪ ∅ = ∅
42	40, 41	eqtr2i 2785	. . 3 ⊢ ∅ = ∪ ran ∅
43		fvprc 6873	. . 3 ⊢ (¬ 𝐴 ∈ V → (TC‘𝐴) = ∅)
44		fvprc 6873	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
45	44	rneqd 5928	. . . 4 ⊢ (¬ 𝐴 ∈ V → ran (𝑈‘𝐴) = ran ∅)
46	45	unieqd 4884	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ran (𝑈‘𝐴) = ∪ ran ∅)
47	42, 43, 46	3eqtr4a 2822	. 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 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  ∪ cuni 4871   ↦ cmpt 5191  Tr wtr 5217  ran crn 5662   ↾ cres 5663  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7861  reccrdg 8395  TCctc 9702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732  ax-inf2 9609

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-tc 9703

This theorem is referenced by:  hsmexlem5  10413