Theorem trclublem 15174

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 15173	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2		ssun1 4179	. . 3 ⊢ 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3		relcnv 6111	. . . . . . . . . . . . . 14 ⊢ Rel ◡𝑅
4		relssdmrn 6286	. . . . . . . . . . . . . 14 ⊢ (Rel ◡𝑅 → ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅))
5	3, 4	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅)
6		ssequn1 4187	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ⊆ (dom ◡𝑅 × ran ◡𝑅) ↔ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅))
7	5, 6	mpbi 233	. . . . . . . . . . . 12 ⊢ (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅)) = (dom ◡𝑅 × ran ◡𝑅)
8		cnvun 6144	. . . . . . . . . . . . 13 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅))
9		cnvxp 6158	. . . . . . . . . . . . . . 15 ⊢ ◡(dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10		df-rn 5666	. . . . . . . . . . . . . . . 16 ⊢ ran 𝑅 = dom ◡𝑅
11		dfdm4 5883	. . . . . . . . . . . . . . . 16 ⊢ dom 𝑅 = ran ◡𝑅
12	10, 11	xpeq12i 5683	. . . . . . . . . . . . . . 15 ⊢ (ran 𝑅 × dom 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
13	9, 12	eqtri 2786	. . . . . . . . . . . . . 14 ⊢ ◡(dom 𝑅 × ran 𝑅) = (dom ◡𝑅 × ran ◡𝑅)
14	13	uneq2i 4167	. . . . . . . . . . . . 13 ⊢ (◡𝑅 ∪ ◡(dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
15	8, 14	eqtri 2786	. . . . . . . . . . . 12 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (◡𝑅 ∪ (dom ◡𝑅 × ran ◡𝑅))
16	7, 15, 13	3eqtr4i 2796	. . . . . . . . . . 11 ⊢ ◡(𝑅 ∪ (dom 𝑅 × ran 𝑅)) = ◡(dom 𝑅 × ran 𝑅)
17	16	breqi 5112	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑏◡(dom 𝑅 × ran 𝑅)𝑎)
18		vex 3458	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19		vex 3458	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
20	18, 19	brcnv 5866	. . . . . . . . . 10 ⊢ (𝑏◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎 ↔ 𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
21	18, 19	brcnv 5866	. . . . . . . . . 10 ⊢ (𝑏◡(dom 𝑅 × ran 𝑅)𝑎 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
22	17, 20, 21	3bitr3i 304	. . . . . . . . 9 ⊢ (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑎(dom 𝑅 × ran 𝑅)𝑏)
23	16	breqi 5112	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑐◡(dom 𝑅 × ran 𝑅)𝑏)
24		vex 3458	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
25	24, 18	brcnv 5866	. . . . . . . . . 10 ⊢ (𝑐◡(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏 ↔ 𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
26	24, 18	brcnv 5866	. . . . . . . . . 10 ⊢ (𝑐◡(dom 𝑅 × ran 𝑅)𝑏 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
27	23, 25, 26	3bitr3i 304	. . . . . . . . 9 ⊢ (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐 ↔ 𝑏(dom 𝑅 × ran 𝑅)𝑐)
28	22, 27	anbi12i 640	. . . . . . . 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 219	. . . . . . 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 1868	. . . . . 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 5529	. . . . 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 5664	. . . . 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 5664	. . . . 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 3964	. . . 4 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35		xptrrel 15159	. . . . 5 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36		ssun2 4180	. . . . 5 ⊢ (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
37	35, 36	sstri 3922	. . . 4 ⊢ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
38	34, 37	sstri 3922	. . 3 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39		trcleq2lem 15170	. . . . 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 3633	. . . 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 251	. . 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 711	. 2 ⊢ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
43	1, 42	syl 18	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152   class class class wbr 5106  {copab 5170   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ∘ ccom 5659  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by:  trclubi  15175  trclubgi  15176  trclub  15177  trclubg  15178