Theorem rclexi 44324

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 4132	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
2		dmun 5902	. . . . . . 7 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
3		dmresi 6056	. . . . . . . 8 ⊢ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
4	3	uneq2i 4120	. . . . . . 7 ⊢ (dom 𝐴 ∪ dom ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
5		ssun1 4132	. . . . . . . 8 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
6		ssequn1 4140	. . . . . . . 8 ⊢ (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
7	5, 6	mpbi 233	. . . . . . 7 ⊢ (dom 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
8	2, 4, 7	3eqtri 2790	. . . . . 6 ⊢ dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
9		rnun 6144	. . . . . . 7 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
10		rnresi 6079	. . . . . . . 8 ⊢ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
11	10	uneq2i 4120	. . . . . . 7 ⊢ (ran 𝐴 ∪ ran ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴))
12		ssun2 4133	. . . . . . . 8 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
13		ssequn1 4140	. . . . . . . 8 ⊢ (ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) ↔ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴))
14	12, 13	mpbi 233	. . . . . . 7 ⊢ (ran 𝐴 ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
15	9, 11, 14	3eqtri 2790	. . . . . 6 ⊢ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) = (dom 𝐴 ∪ ran 𝐴)
16	8, 15	uneq12i 4121	. . . . 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 4112	. . . . 5 ⊢ ((dom 𝐴 ∪ ran 𝐴) ∪ (dom 𝐴 ∪ ran 𝐴)) = (dom 𝐴 ∪ ran 𝐴)
18	16, 17	eqtri 2786	. . . 4 ⊢ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))) = (dom 𝐴 ∪ ran 𝐴)
19	18	reseq2i 5977	. . 3 ⊢ ( I ↾ (dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∪ ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))) = ( I ↾ (dom 𝐴 ∪ ran 𝐴))
20		ssun2 4133	. . 3 ⊢ ( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
21	19, 20	eqsstri 3984	. 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 3477	. . . . 5 ⊢ 𝐴 ∈ V
24		dmexg 7899	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V)
25		rnexg 7900	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
26	24, 25	unexd 7754	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (dom 𝐴 ∪ ran 𝐴) ∈ V)
27	26	resiexd 7216	. . . . . 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 7744	. . . 4 ⊢ (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) ∈ V
30		dmeq 5895	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → dom 𝑥 = dom (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
31		rneq 5928	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → ran 𝑥 = ran (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
32	30, 31	uneq12d 4124	. . . . . . 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 5980	. . . . . 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 23	. . . . . 6 ⊢ (𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))) → 𝑥 = (𝐴 ∪ ( I ↾ (dom 𝐴 ∪ ran 𝐴))))
35	33, 34	sseq12d 3971	. . . . 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 44317	. . . 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 3566	. . 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 5318	. . 3 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V)
39	37, 38	sylib 221	. 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 704	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455   ∪ cun 3904   ⊆ wss 3906  ∩ cint 4913   I cid 5557  dom cdm 5663  ran crn 5664   ↾ cres 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is referenced by: (None)