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Theorem trclublem 14978
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 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 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.
StepHypRef Expression
1 trclexlem 14977 . 2 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2 ssun1 4170 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3 relcnv 6109 . . . . . . . . . . . . . 14 Rel 𝑅
4 relssdmrn 6274 . . . . . . . . . . . . . 14 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4ax-mp 5 . . . . . . . . . . . . 13 𝑅 ⊆ (dom 𝑅 × ran 𝑅)
6 ssequn1 4178 . . . . . . . . . . . . 13 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
75, 6mpbi 229 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
8 cnvun 6149 . . . . . . . . . . . . 13 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅(dom 𝑅 × ran 𝑅))
9 cnvxp 6163 . . . . . . . . . . . . . . 15 (dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10 df-rn 5689 . . . . . . . . . . . . . . . 16 ran 𝑅 = dom 𝑅
11 dfdm4 5898 . . . . . . . . . . . . . . . 16 dom 𝑅 = ran 𝑅
1210, 11xpeq12i 5706 . . . . . . . . . . . . . . 15 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
139, 12eqtri 2753 . . . . . . . . . . . . . 14 (dom 𝑅 × ran 𝑅) = (dom 𝑅 × ran 𝑅)
1413uneq2i 4157 . . . . . . . . . . . . 13 (𝑅(dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
158, 14eqtri 2753 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
167, 15, 133eqtr4i 2763 . . . . . . . . . . 11 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
1716breqi 5155 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑏(dom 𝑅 × ran 𝑅)𝑎)
18 vex 3465 . . . . . . . . . . 11 𝑏 ∈ V
19 vex 3465 . . . . . . . . . . 11 𝑎 ∈ V
2018, 19brcnv 5885 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
2118, 19brcnv 5885 . . . . . . . . . 10 (𝑏(dom 𝑅 × ran 𝑅)𝑎𝑎(dom 𝑅 × ran 𝑅)𝑏)
2217, 20, 213bitr3i 300 . . . . . . . . 9 (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑎(dom 𝑅 × ran 𝑅)𝑏)
2316breqi 5155 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑐(dom 𝑅 × ran 𝑅)𝑏)
24 vex 3465 . . . . . . . . . . 11 𝑐 ∈ V
2524, 18brcnv 5885 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
2624, 18brcnv 5885 . . . . . . . . . 10 (𝑐(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐)
2723, 25, 263bitr3i 300 . . . . . . . . 9 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐𝑏(dom 𝑅 × ran 𝑅)𝑐)
2822, 27anbi12i 626 . . . . . . . 8 ((𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐) ↔ (𝑎(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐))
2928biimpi 215 . . . . . . 7 ((𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐) → (𝑎(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐))
3029eximi 1829 . . . . . 6 (∃𝑏(𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐) → ∃𝑏(𝑎(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐))
3130ssopab2i 5552 . . . . 5 {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)} ⊆ {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐)}
32 df-co 5687 . . . . 5 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)}
33 df-co 5687 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑏(𝑎(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐)}
3431, 32, 333sstr4i 4020 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35 xptrrel 14963 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36 ssun2 4171 . . . . 5 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3735, 36sstri 3986 . . . 4 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3834, 37sstri 3986 . . 3 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39 trcleq2lem 14974 . . . . 5 (𝑥 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)) → ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∧ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))))
4039elabg 3662 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V → ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ↔ (𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∧ ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))))
4140biimprd 247 . . 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 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}))
422, 38, 41mp2ani 696 . 2 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
431, 42syl 17 1 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  {cab 2702  Vcvv 3461  cun 3942  wss 3944   class class class wbr 5149  {copab 5211   × cxp 5676  ccnv 5677  dom cdm 5678  ran crn 5679  ccom 5682  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690
This theorem is referenced by:  trclubi  14979  trclubgi  14980  trclub  14981  trclubg  14982
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