Theorem trclexi 43591

Description: The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)

Hypothesis

Ref	Expression
trclexi.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
trclexi	⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trclexi

Step	Hyp	Ref	Expression
1		ssun1 4153	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2		coundir 6237	. . . 4 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))))
3		coundi 6236	. . . . . 6 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4		cossxp 6261	. . . . . . 7 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5		cossxp 6261	. . . . . . . 8 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6		dmxpss 6160	. . . . . . . . 9 ⊢ dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7		xpss1 5673	. . . . . . . . 9 ⊢ (dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴 → (dom (dom 𝐴 × ran 𝐴) × ran 𝐴) ⊆ (dom 𝐴 × ran 𝐴))
8	6, 7	ax-mp 5	. . . . . . . 8 ⊢ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
9	5, 8	sstri 3968	. . . . . . 7 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
10	4, 9	unssi 4166	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
11	3, 10	eqsstri 4005	. . . . 5 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12		coundi 6236	. . . . . 6 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)))
13		cossxp 6261	. . . . . . . 8 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14		rnxpss 6161	. . . . . . . . 9 ⊢ ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15		xpss2 5674	. . . . . . . . 9 ⊢ (ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴 → (dom 𝐴 × ran (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴))
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
17	13, 16	sstri 3968	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18		xptrrel 14997	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
19	17, 18	unssi 4166	. . . . . 6 ⊢ (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
20	12, 19	eqsstri 4005	. . . . 5 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
21	11, 20	unssi 4166	. . . 4 ⊢ ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
22	2, 21	eqsstri 4005	. . 3 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23		ssun2 4154	. . 3 ⊢ (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
24	22, 23	sstri 3968	. 2 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
25		trclexi.1	. . . . . 6 ⊢ 𝐴 ∈ 𝑉
26	25	elexi 3482	. . . . 5 ⊢ 𝐴 ∈ V
27	26	dmex 7903	. . . . . 6 ⊢ dom 𝐴 ∈ V
28	26	rnex 7904	. . . . . 6 ⊢ ran 𝐴 ∈ V
29	27, 28	xpex 7745	. . . . 5 ⊢ (dom 𝐴 × ran 𝐴) ∈ V
30	26, 29	unex 7736	. . . 4 ⊢ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31		trcleq2lem 15008	. . . 4 ⊢ (𝑥 = (𝐴 ∪ (dom 𝐴 × ran 𝐴)) → ((𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ (𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∧ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))))
32	30, 31	spcev 3585	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∧ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥))
33		intexab 5316	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
34	32, 33	sylib 218	. 2 ⊢ ((𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∧ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
35	1, 24, 34	mp2an 692	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 395  ∃wex 1779   ∈ wcel 2108  {cab 2713  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  ∩ cint 4922   × cxp 5652  dom cdm 5654  ran crn 5655   ∘ ccom 5658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666

This theorem is referenced by:  dfrtrcl5  43600