Theorem trclublem 14357

Description: If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.)

Assertion

Ref	Expression
trclublem	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Distinct variable group:   𝑥,𝑅

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trclublem

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trclexlem 14356	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2		ssun1 4150	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3		relcnv 5969	. . . . . . . . . . . . . 14 ⊢ Rel ◡𝑅
4		relssdmrn 6123	. . . . . . . . . . . . . 14 ⊢ (Rel ◡𝑅 → ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅))
5	3, 4	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅)
6		ssequn1 4158	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅) ↔ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅))
7	5, 6	mpbi 232	. . . . . . . . . . . 12 ⊢ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅)
8		cnvun 6003	. . . . . . . . . . . . 13 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅))
9		cnvxp 6016	. . . . . . . . . . . . . . 15 ⊢ ◡(dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10		df-rn 5568	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑅 = dom ◡𝑅
11		dfdm4 5766	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅 = ran ◡𝑅
12	10, 11	xpeq12i 5585	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑅 × dom 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
13	9, 12	eqtri 2846	. . . . . . . . . . . . . 14 ⊢ ◡(dom 𝑅 × ran 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
14	13	uneq2i 4138	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
15	8, 14	eqtri 2846	. . . . . . . . . . . 12 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
16	7, 15, 13	3eqtr4i 2856	. . . . . . . . . . 11 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ◡(dom 𝑅 × ran 𝑅)
17	16	breqi 5074	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑏◡(dom 𝑅 × ran 𝑅)𝑎)
18		vex 3499	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19		vex 3499	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
20	18, 19	brcnv 5755	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
21	18, 19	brcnv 5755	. . . . . . . . . 10 ⊢ (𝑏◡(dom 𝑅 × ran 𝑅)𝑎 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
22	17, 20, 21	3bitr3i 303	. . . . . . . . 9 ⊢ (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
23	16	breqi 5074	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑐◡(dom 𝑅 × ran 𝑅)𝑏)
24		vex 3499	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
25	24, 18	brcnv 5755	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
26	24, 18	brcnv 5755	. . . . . . . . . 10 ⊢ (𝑐◡(dom 𝑅 × ran 𝑅)𝑏 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
27	23, 25, 26	3bitr3i 303	. . . . . . . . 9 ⊢ (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
28	22, 27	anbi12i 628	. . . . . . . 8 ⊢ ((𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ∧ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐) ↔ (𝑎(dom 𝑅 × ran 𝑅)𝑏 ∧ 𝑏(dom 𝑅 × ran 𝑅)𝑐))
29	28	biimpi 218	. . . . . . 7 ⊢ ((𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ∧ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐) → (𝑎(dom 𝑅 × ran 𝑅)𝑏 ∧ 𝑏(dom 𝑅 × ran 𝑅)𝑐))
30	29	eximi 1835	. . . . . 6 ⊢ (∃𝑏(𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ∧ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐) → ∃𝑏(𝑎(dom 𝑅 × ran 𝑅)𝑏 ∧ 𝑏(dom 𝑅 × ran 𝑅)𝑐))
31	30	ssopab2i 5439	. . . . 5 ⊢ {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ∧ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)} ⊆ {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(dom 𝑅 × ran 𝑅)𝑏 ∧ 𝑏(dom 𝑅 × ran 𝑅)𝑐)}
32		df-co 5566	. . . . 5 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ∧ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)}
33		df-co 5566	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(dom 𝑅 × ran 𝑅)𝑏 ∧ 𝑏(dom 𝑅 × ran 𝑅)𝑐)}
34	31, 32, 33	3sstr4i 4012	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35		xptrrel 14342	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36		ssun2 4151	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
37	35, 36	sstri 3978	. . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
38	34, 37	sstri 3978	. . 3 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39		trcleq2lem 14353	. . . . 5 ⊢ (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ((𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥) ↔ (𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∧ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))))
40	39	elabg 3668	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ↔ (𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∧ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))))
41	40	biimprd 250	. . 3 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V → ((𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∧ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}))
42	2, 38, 41	mp2ani 696	. 2 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
43	1, 42	syl 17	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801  Vcvv 3496   ∪ cun 3936   ⊆ wss 3938   class class class wbr 5068  {copab 5130   × cxp 5555  ^◡ccnv 5556  dom cdm 5557  ran crn 5558   ∘ ccom 5561  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569

This theorem is referenced by:  trclubi  14358  trclubgi  14359  trclub  14360  trclubg  14361