Theorem rclexi 42414

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 4173	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
2		dmun 5911	. . . . . . 7 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
3		dmresi 6052	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
4	3	uneq2i 4161	. . . . . . 7 ⊢ (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
5		ssun1 4173	. . . . . . . 8 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		ssequn1 4181	. . . . . . . 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 2765	. . . . . 6 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
9		rnun 6146	. . . . . . 7 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
10		rnresi 6075	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
11	10	uneq2i 4161	. . . . . . 7 ⊢ (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
12		ssun2 4174	. . . . . . . 8 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
13		ssequn1 4181	. . . . . . . 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 2765	. . . . . 6 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
16	8, 15	uneq12i 4162	. . . . 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 4153	. . . . 5 ⊢ ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
18	16, 17	eqtri 2761	. . . 4 ⊢ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = (dom 𝐴 ∪ ran 𝐴)
19	18	reseq2i 5979	. . 3 ⊢ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
20		ssun2 4174	. . 3 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
21	19, 20	eqsstri 4017	. 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 3494	. . . . 5 ⊢ 𝐴 ∈ V
24		dmexg 7894	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
25		rnexg 7895	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
26	24, 25	unexd 7741	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (dom 𝐴 ∪ ran 𝐴) ∈ V)
27	26	resiexd 7218	. . . . . 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 7733	. . . 4 ⊢ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∈ V
30		dmeq 5904	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → dom 𝑥 = dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
31		rneq 5936	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
32	30, 31	uneq12d 4165	. . . . . . 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 5982	. . . . . 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 4016	. . . . 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 42407	. . . 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 3597	. . 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 5340	. . 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 691	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  Vcvv 3475   ∪ cun 3947   ⊆ wss 3949  ∩ cint 4951   I cid 5574  dom cdm 5677  ran crn 5678   ↾ cres 5679

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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552

This theorem is referenced by: (None)