Theorem trclublem 15034

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 15033	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2		ssun1 4178	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3		relcnv 6122	. . . . . . . . . . . . . 14 ⊢ Rel ◡𝑅
4		relssdmrn 6288	. . . . . . . . . . . . . 14 ⊢ (Rel ◡𝑅 → ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅))
5	3, 4	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅)
6		ssequn1 4186	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅) ↔ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅))
7	5, 6	mpbi 230	. . . . . . . . . . . 12 ⊢ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅)
8		cnvun 6162	. . . . . . . . . . . . 13 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅))
9		cnvxp 6177	. . . . . . . . . . . . . . 15 ⊢ ◡(dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10		df-rn 5696	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑅 = dom ◡𝑅
11		dfdm4 5906	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅 = ran ◡𝑅
12	10, 11	xpeq12i 5713	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑅 × dom 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
13	9, 12	eqtri 2765	. . . . . . . . . . . . . 14 ⊢ ◡(dom 𝑅 × ran 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
14	13	uneq2i 4165	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
15	8, 14	eqtri 2765	. . . . . . . . . . . 12 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
16	7, 15, 13	3eqtr4i 2775	. . . . . . . . . . 11 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ◡(dom 𝑅 × ran 𝑅)
17	16	breqi 5149	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑏◡(dom 𝑅 × ran 𝑅)𝑎)
18		vex 3484	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19		vex 3484	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
20	18, 19	brcnv 5893	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
21	18, 19	brcnv 5893	. . . . . . . . . 10 ⊢ (𝑏◡(dom 𝑅 × ran 𝑅)𝑎 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
22	17, 20, 21	3bitr3i 301	. . . . . . . . 9 ⊢ (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
23	16	breqi 5149	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑐◡(dom 𝑅 × ran 𝑅)𝑏)
24		vex 3484	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
25	24, 18	brcnv 5893	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
26	24, 18	brcnv 5893	. . . . . . . . . 10 ⊢ (𝑐◡(dom 𝑅 × ran 𝑅)𝑏 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
27	23, 25, 26	3bitr3i 301	. . . . . . . . 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 216	. . . . . . 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 5555	. . . . 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 5694	. . . . 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 5694	. . . . 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 4035	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35		xptrrel 15019	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36		ssun2 4179	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
37	35, 36	sstri 3993	. . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
38	34, 37	sstri 3993	. . 3 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39		trcleq2lem 15030	. . . . 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 3676	. . . 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 248	. . 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 698	. 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 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951   class class class wbr 5143  {copab 5205   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ∘ ccom 5689  Rel wrel 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697

This theorem is referenced by:  trclubi  15035  trclubgi  15036  trclub  15037  trclubg  15038