Theorem rclexi 43577

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

Hypothesis

Ref	Expression
rclexi.1	⊢ 𝐴 ∈ 𝑉

Assertion

Ref	Expression
rclexi	⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rclexi

Step	Hyp	Ref	Expression
1		ssun1 4137	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
2		dmun 5864	. . . . . . 7 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
3		dmresi 6012	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
4	3	uneq2i 4124	. . . . . . 7 ⊢ (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
5		ssun1 4137	. . . . . . . 8 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		ssequn1 4145	. . . . . . . 8 ⊢ (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
7	5, 6	mpbi 230	. . . . . . 7 ⊢ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
8	2, 4, 7	3eqtri 2756	. . . . . 6 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
9		rnun 6106	. . . . . . 7 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
10		rnresi 6035	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
11	10	uneq2i 4124	. . . . . . 7 ⊢ (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
12		ssun2 4138	. . . . . . . 8 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
13		ssequn1 4145	. . . . . . . 8 ⊢ (ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
14	12, 13	mpbi 230	. . . . . . 7 ⊢ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
15	9, 11, 14	3eqtri 2756	. . . . . 6 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
16	8, 15	uneq12i 4125	. . . . 5 ⊢ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴))
17		unidm 4116	. . . . 5 ⊢ ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
18	16, 17	eqtri 2752	. . . 4 ⊢ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = (dom 𝐴 ∪ ran 𝐴)
19	18	reseq2i 5936	. . 3 ⊢ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
20		ssun2 4138	. . 3 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
21	19, 20	eqsstri 3990	. 2 ⊢ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
22		rclexi.1	. . . . . 6 ⊢ 𝐴 ∈ 𝑉
23	22	elexi 3467	. . . . 5 ⊢ 𝐴 ∈ V
24		dmexg 7857	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
25		rnexg 7858	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
26	24, 25	unexd 7710	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (dom 𝐴 ∪ ran 𝐴) ∈ V)
27	26	resiexd 7172	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ∈ V)
28	22, 27	ax-mp 5	. . . . 5 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ∈ V
29	23, 28	unex 7700	. . . 4 ⊢ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∈ V
30		dmeq 5857	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → dom 𝑥 = dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
31		rneq 5889	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
32	30, 31	uneq12d 4128	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → (dom 𝑥 ∪ ran 𝑥) = (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))))
33	32	reseq2d 5939	. . . . . 6 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))))
34		id 22	. . . . . 6 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → 𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
35	33, 34	sseq12d 3977	. . . . 5 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥 ↔ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))))
36	35	cleq2lem 43570	. . . 4 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ((𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ (𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))))
37	29, 36	spcev 3569	. . 3 ⊢ ((𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))
38		intexab 5296	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V)
39	37, 38	sylib 218	. 2 ⊢ ((𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∧ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V)
40	1, 21, 39	mp2an 692	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3444   ∪ cun 3909   ⊆ wss 3911  ∩ cint 4906   I cid 5525  dom cdm 5631  ran crn 5632   ↾ cres 5633

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by: (None)