Theorem trclexi 41441

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 4112	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2		coundir 6166	. . . 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 6165	. . . . . 6 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4		cossxp 6190	. . . . . . 7 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5		cossxp 6190	. . . . . . . 8 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6		dmxpss 6089	. . . . . . . . 9 ⊢ dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7		xpss1 5619	. . . . . . . . 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 3935	. . . . . . 7 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
10	4, 9	unssi 4125	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
11	3, 10	eqsstri 3960	. . . . 5 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12		coundi 6165	. . . . . 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 6190	. . . . . . . 8 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14		rnxpss 6090	. . . . . . . . 9 ⊢ ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15		xpss2 5620	. . . . . . . . 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 3935	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18		xptrrel 14740	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
19	17, 18	unssi 4125	. . . . . 6 ⊢ (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
20	12, 19	eqsstri 3960	. . . . 5 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
21	11, 20	unssi 4125	. . . 4 ⊢ ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
22	2, 21	eqsstri 3960	. . 3 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23		ssun2 4113	. . 3 ⊢ (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
24	22, 23	sstri 3935	. 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 3456	. . . . 5 ⊢ 𝐴 ∈ V
27	26	dmex 7790	. . . . . 6 ⊢ dom 𝐴 ∈ V
28	26	rnex 7791	. . . . . 6 ⊢ ran 𝐴 ∈ V
29	27, 28	xpex 7635	. . . . 5 ⊢ (dom 𝐴 × ran 𝐴) ∈ V
30	26, 29	unex 7628	. . . 4 ⊢ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31		trcleq2lem 14751	. . . 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 3550	. . 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 5272	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
34	32, 33	sylib 217	. 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 690	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 397  ∃wex 1779   ∈ wcel 2104  {cab 2713  Vcvv 3437   ∪ cun 3890   ⊆ wss 3892  ∩ cint 4886   × cxp 5598  dom cdm 5600  ran crn 5601   ∘ ccom 5604

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-br 5082  df-opab 5144  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612

This theorem is referenced by:  dfrtrcl5  41450