Theorem trclexi 42357

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 4172	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2		coundir 6245	. . . 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 6244	. . . . . 6 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4		cossxp 6269	. . . . . . 7 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5		cossxp 6269	. . . . . . . 8 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6		dmxpss 6168	. . . . . . . . 9 ⊢ dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7		xpss1 5695	. . . . . . . . 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 3991	. . . . . . 7 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
10	4, 9	unssi 4185	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
11	3, 10	eqsstri 4016	. . . . 5 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12		coundi 6244	. . . . . 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 6269	. . . . . . . 8 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14		rnxpss 6169	. . . . . . . . 9 ⊢ ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15		xpss2 5696	. . . . . . . . 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 3991	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18		xptrrel 14924	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
19	17, 18	unssi 4185	. . . . . 6 ⊢ (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
20	12, 19	eqsstri 4016	. . . . 5 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
21	11, 20	unssi 4185	. . . 4 ⊢ ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
22	2, 21	eqsstri 4016	. . 3 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23		ssun2 4173	. . 3 ⊢ (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
24	22, 23	sstri 3991	. 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 3494	. . . . 5 ⊢ 𝐴 ∈ V
27	26	dmex 7899	. . . . . 6 ⊢ dom 𝐴 ∈ V
28	26	rnex 7900	. . . . . 6 ⊢ ran 𝐴 ∈ V
29	27, 28	xpex 7737	. . . . 5 ⊢ (dom 𝐴 × ran 𝐴) ∈ V
30	26, 29	unex 7730	. . . 4 ⊢ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31		trcleq2lem 14935	. . . 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 3597	. . 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 5339	. . 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 691	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 397  ∃wex 1782   ∈ wcel 2107  {cab 2710  Vcvv 3475   ∪ cun 3946   ⊆ wss 3948  ∩ cint 4950   × cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688

This theorem is referenced by:  dfrtrcl5  42366