Theorem trclexi 44346

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 4131	. 2 ⊢ 𝐴 ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
2		coundir 6249	. . . 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 6248	. . . . . 6 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) = ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴)))
4		cossxp 6273	. . . . . . 7 ⊢ (𝐴 ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
5		cossxp 6273	. . . . . . . 8 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom (dom 𝐴 × ran 𝐴) × ran 𝐴)
6		dmxpss 6169	. . . . . . . . 9 ⊢ dom (dom 𝐴 × ran 𝐴) ⊆ dom 𝐴
7		xpss1 5680	. . . . . . . . 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 3946	. . . . . . 7 ⊢ (𝐴 ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
10	4, 9	unssi 4144	. . . . . 6 ⊢ ((𝐴 ∘ 𝐴) ∪ (𝐴 ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
11	3, 10	eqsstri 3983	. . . . 5 ⊢ (𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
12		coundi 6248	. . . . . 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 6273	. . . . . . . 8 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran (dom 𝐴 × ran 𝐴))
14		rnxpss 6170	. . . . . . . . 9 ⊢ ran (dom 𝐴 × ran 𝐴) ⊆ ran 𝐴
15		xpss2 5681	. . . . . . . . 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 3946	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ 𝐴) ⊆ (dom 𝐴 × ran 𝐴)
18		xptrrel 15013	. . . . . . 7 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴)) ⊆ (dom 𝐴 × ran 𝐴)
19	17, 18	unssi 4144	. . . . . 6 ⊢ (((dom 𝐴 × ran 𝐴) ∘ 𝐴) ∪ ((dom 𝐴 × ran 𝐴) ∘ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
20	12, 19	eqsstri 3983	. . . . 5 ⊢ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
21	11, 20	unssi 4144	. . . 4 ⊢ ((𝐴 ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ∪ ((dom 𝐴 × ran 𝐴) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴)))) ⊆ (dom 𝐴 × ran 𝐴)
22	2, 21	eqsstri 3983	. . 3 ⊢ ((𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∘ (𝐴 ∪ (dom 𝐴 × ran 𝐴))) ⊆ (dom 𝐴 × ran 𝐴)
23		ssun2 4132	. . 3 ⊢ (dom 𝐴 × ran 𝐴) ⊆ (𝐴 ∪ (dom 𝐴 × ran 𝐴))
24	22, 23	sstri 3946	. 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 3477	. . . . 5 ⊢ 𝐴 ∈ V
27	26	dmex 7902	. . . . . 6 ⊢ dom 𝐴 ∈ V
28	26	rnex 7903	. . . . . 6 ⊢ ran 𝐴 ∈ V
29	27, 28	xpex 7748	. . . . 5 ⊢ (dom 𝐴 × ran 𝐴) ∈ V
30	26, 29	unex 7742	. . . 4 ⊢ (𝐴 ∪ (dom 𝐴 × ran 𝐴)) ∈ V
31		trcleq2lem 15024	. . . 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 3565	. . 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 5316	. . 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 704	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V

Syntax hints:   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  {cab 2741  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  ∩ cint 4912   × cxp 5659  dom cdm 5661  ran crn 5662   ∘ ccom 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-sep 5257  ax-pow 5336  ax-pr 5404  ax-un 7732

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-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673

This theorem is referenced by:  dfrtrcl5  44355