Theorem trclublem 13734

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 13733	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2		ssun1 3776	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3		relcnv 5503	. . . . . . . . . . . . . 14 ⊢ Rel ◡𝑅
4		relssdmrn 5656	. . . . . . . . . . . . . 14 ⊢ (Rel ◡𝑅 → ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅))
5	3, 4	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅)
6		ssequn1 3783	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅) ↔ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅))
7	5, 6	mpbi 220	. . . . . . . . . . . 12 ⊢ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅)
8		cnvun 5538	. . . . . . . . . . . . 13 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅))
9		cnvxp 5551	. . . . . . . . . . . . . . 15 ⊢ ◡(dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10		df-rn 5125	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑅 = dom ◡𝑅
11		dfdm4 5316	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅 = ran ◡𝑅
12	10, 11	xpeq12i 5137	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑅 × dom 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
13	9, 12	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ ◡(dom 𝑅 × ran 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
14	13	uneq2i 3764	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
15	8, 14	eqtri 2644	. . . . . . . . . . . 12 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
16	7, 15, 13	3eqtr4i 2654	. . . . . . . . . . 11 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ◡(dom 𝑅 × ran 𝑅)
17	16	breqi 4659	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑏◡(dom 𝑅 × ran 𝑅)𝑎)
18		vex 3203	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19		vex 3203	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
20	18, 19	brcnv 5305	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
21	18, 19	brcnv 5305	. . . . . . . . . 10 ⊢ (𝑏◡(dom 𝑅 × ran 𝑅)𝑎 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
22	17, 20, 21	3bitr3i 290	. . . . . . . . 9 ⊢ (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
23	16	breqi 4659	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑐◡(dom 𝑅 × ran 𝑅)𝑏)
24		vex 3203	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
25	24, 18	brcnv 5305	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
26	24, 18	brcnv 5305	. . . . . . . . . 10 ⊢ (𝑐◡(dom 𝑅 × ran 𝑅)𝑏 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
27	23, 25, 26	3bitr3i 290	. . . . . . . . 9 ⊢ (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
28	22, 27	anbi12i 733	. . . . . . . 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 206	. . . . . . 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 1762	. . . . . 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 5003	. . . . 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 5123	. . . . 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 5123	. . . . 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 3644	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35		xptrrel 13719	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36		ssun2 3777	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
37	35, 36	sstri 3612	. . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
38	34, 37	sstri 3612	. . 3 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39		trcleq2lem 13730	. . . . 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 3351	. . . 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 238	. . 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 714	. 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 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574   class class class wbr 4653  {copab 4712   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ∘ ccom 5118  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  trclubi  13735  trclubiOLD  13736  trclubgi  13737  trclubgiOLD  13738  trclub  13739  trclubg  13740