Theorem rclexi 43353

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 4183	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
2		dmun 5921	. . . . . . 7 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
3		dmresi 6066	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
4	3	uneq2i 4170	. . . . . . 7 ⊢ (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
5		ssun1 4183	. . . . . . . 8 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		ssequn1 4191	. . . . . . . 8 ⊢ (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
7	5, 6	mpbi 229	. . . . . . 7 ⊢ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
8	2, 4, 7	3eqtri 2761	. . . . . 6 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
9		rnun 6162	. . . . . . 7 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
10		rnresi 6089	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
11	10	uneq2i 4170	. . . . . . 7 ⊢ (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
12		ssun2 4184	. . . . . . . 8 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
13		ssequn1 4191	. . . . . . . 8 ⊢ (ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
14	12, 13	mpbi 229	. . . . . . 7 ⊢ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
15	9, 11, 14	3eqtri 2761	. . . . . 6 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
16	8, 15	uneq12i 4171	. . . . 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 4162	. . . . 5 ⊢ ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
18	16, 17	eqtri 2757	. . . 4 ⊢ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = (dom 𝐴 ∪ ran 𝐴)
19	18	reseq2i 5991	. . 3 ⊢ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
20		ssun2 4184	. . 3 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
21	19, 20	eqsstri 4026	. 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 3495	. . . . 5 ⊢ 𝐴 ∈ V
24		dmexg 7922	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
25		rnexg 7923	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
26	24, 25	unexd 7770	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (dom 𝐴 ∪ ran 𝐴) ∈ V)
27	26	resiexd 7238	. . . . . 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 7760	. . . 4 ⊢ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∈ V
30		dmeq 5914	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → dom 𝑥 = dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
31		rneq 5946	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
32	30, 31	uneq12d 4174	. . . . . . 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 5994	. . . . . 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 4025	. . . . 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 43346	. . . 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 3604	. . 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 5354	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V)
39	37, 38	sylib 217	. 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 690	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2100  {cab 2706  Vcvv 3472   ∪ cun 3957   ⊆ wss 3959  ∩ cint 4959   I cid 5583  dom cdm 5686  ran crn 5687   ↾ cres 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5293  ax-sep 5307  ax-nul 5314  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3789  df-csb 3905  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-op 4643  df-uni 4919  df-int 4960  df-iun 5008  df-br 5157  df-opab 5219  df-mpt 5240  df-id 5584  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-rn 5697  df-res 5698  df-ima 5699  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566

This theorem is referenced by: (None)