MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trclublem Structured version   Visualization version   GIF version

Theorem trclublem 14143
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 14142 . 2 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2 ssun1 3999 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3 relcnv 5757 . . . . . . . . . . . . . 14 Rel 𝑅
4 relssdmrn 5910 . . . . . . . . . . . . . 14 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4ax-mp 5 . . . . . . . . . . . . 13 𝑅 ⊆ (dom 𝑅 × ran 𝑅)
6 ssequn1 4006 . . . . . . . . . . . . 13 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
75, 6mpbi 222 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
8 cnvun 5792 . . . . . . . . . . . . 13 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅(dom 𝑅 × ran 𝑅))
9 cnvxp 5805 . . . . . . . . . . . . . . 15 (dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10 df-rn 5366 . . . . . . . . . . . . . . . 16 ran 𝑅 = dom 𝑅
11 dfdm4 5561 . . . . . . . . . . . . . . . 16 dom 𝑅 = ran 𝑅
1210, 11xpeq12i 5383 . . . . . . . . . . . . . . 15 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
139, 12eqtri 2802 . . . . . . . . . . . . . 14 (dom 𝑅 × ran 𝑅) = (dom 𝑅 × ran 𝑅)
1413uneq2i 3987 . . . . . . . . . . . . 13 (𝑅(dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
158, 14eqtri 2802 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
167, 15, 133eqtr4i 2812 . . . . . . . . . . 11 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
1716breqi 4892 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑏(dom 𝑅 × ran 𝑅)𝑎)
18 vex 3401 . . . . . . . . . . 11 𝑏 ∈ V
19 vex 3401 . . . . . . . . . . 11 𝑎 ∈ V
2018, 19brcnv 5550 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
2118, 19brcnv 5550 . . . . . . . . . 10 (𝑏(dom 𝑅 × ran 𝑅)𝑎𝑎(dom 𝑅 × ran 𝑅)𝑏)
2217, 20, 213bitr3i 293 . . . . . . . . 9 (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑎(dom 𝑅 × ran 𝑅)𝑏)
2316breqi 4892 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑐(dom 𝑅 × ran 𝑅)𝑏)
24 vex 3401 . . . . . . . . . . 11 𝑐 ∈ V
2524, 18brcnv 5550 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
2624, 18brcnv 5550 . . . . . . . . . 10 (𝑐(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐)
2723, 25, 263bitr3i 293 . . . . . . . . 9 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐𝑏(dom 𝑅 × ran 𝑅)𝑐)
2822, 27anbi12i 620 . . . . . . . 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 208 . . . . . . 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 1878 . . . . . 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 5240 . . . . 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 5364 . . . . 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 5364 . . . . 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 3863 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35 xptrrel 14128 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36 ssun2 4000 . . . . 5 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3735, 36sstri 3830 . . . 4 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3834, 37sstri 3830 . . 3 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39 trcleq2lem 14139 . . . . 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 3556 . . . 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 240 . . 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 688 . 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 386   = wceq 1601  wex 1823  wcel 2107  {cab 2763  Vcvv 3398  cun 3790  wss 3792   class class class wbr 4886  {copab 4948   × cxp 5353  ccnv 5354  dom cdm 5355  ran crn 5356  ccom 5359  Rel wrel 5360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367
This theorem is referenced by:  trclubi  14144  trclubgi  14145  trclub  14146  trclubg  14147
  Copyright terms: Public domain W3C validator