Theorem trclexi 40320

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 4099	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2		coundir 6068	. . . 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 6067	. . . . . 6 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4		cossxp 6091	. . . . . . 7 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5		cossxp 6091	. . . . . . . 8 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6		dmxpss 5995	. . . . . . . . 9 ⊢ dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7		xpss1 5538	. . . . . . . . 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 3924	. . . . . . 7 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
10	4, 9	unssi 4112	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
11	3, 10	eqsstri 3949	. . . . 5 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12		coundi 6067	. . . . . 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 6091	. . . . . . . 8 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14		rnxpss 5996	. . . . . . . . 9 ⊢ ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15		xpss2 5539	. . . . . . . . 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 3924	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18		xptrrel 14331	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
19	17, 18	unssi 4112	. . . . . 6 ⊢ (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
20	12, 19	eqsstri 3949	. . . . 5 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
21	11, 20	unssi 4112	. . . 4 ⊢ ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
22	2, 21	eqsstri 3949	. . 3 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23		ssun2 4100	. . 3 ⊢ (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
24	22, 23	sstri 3924	. 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 3460	. . . . 5 ⊢ 𝐴 ∈ V
27	26	dmex 7598	. . . . . 6 ⊢ dom 𝐴 ∈ V
28	26	rnex 7599	. . . . . 6 ⊢ ran 𝐴 ∈ V
29	27, 28	xpex 7456	. . . . 5 ⊢ (dom 𝐴 × ran 𝐴) ∈ V
30	26, 29	unex 7449	. . . 4 ⊢ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31		trcleq2lem 14342	. . . 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 3555	. . 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 5206	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V)
34	32, 33	sylib 221	. 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 399  ∃wex 1781   ∈ wcel 2111  {cab 2776  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ∩ cint 4838   × cxp 5517  dom cdm 5519  ran crn 5520   ∘ ccom 5523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531

This theorem is referenced by:  dfrtrcl5  40329