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Theorem trclublem 13780
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 13779 . 2 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2 ssun1 3809 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3 relcnv 5538 . . . . . . . . . . . . . 14 Rel 𝑅
4 relssdmrn 5694 . . . . . . . . . . . . . 14 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4ax-mp 5 . . . . . . . . . . . . 13 𝑅 ⊆ (dom 𝑅 × ran 𝑅)
6 ssequn1 3816 . . . . . . . . . . . . 13 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
75, 6mpbi 220 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
8 cnvun 5573 . . . . . . . . . . . . 13 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅(dom 𝑅 × ran 𝑅))
9 cnvxp 5586 . . . . . . . . . . . . . . 15 (dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10 df-rn 5154 . . . . . . . . . . . . . . . 16 ran 𝑅 = dom 𝑅
11 dfdm4 5348 . . . . . . . . . . . . . . . 16 dom 𝑅 = ran 𝑅
1210, 11xpeq12i 5171 . . . . . . . . . . . . . . 15 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
139, 12eqtri 2673 . . . . . . . . . . . . . 14 (dom 𝑅 × ran 𝑅) = (dom 𝑅 × ran 𝑅)
1413uneq2i 3797 . . . . . . . . . . . . 13 (𝑅(dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
158, 14eqtri 2673 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
167, 15, 133eqtr4i 2683 . . . . . . . . . . 11 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
1716breqi 4691 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑏(dom 𝑅 × ran 𝑅)𝑎)
18 vex 3234 . . . . . . . . . . 11 𝑏 ∈ V
19 vex 3234 . . . . . . . . . . 11 𝑎 ∈ V
2018, 19brcnv 5337 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
2118, 19brcnv 5337 . . . . . . . . . 10 (𝑏(dom 𝑅 × ran 𝑅)𝑎𝑎(dom 𝑅 × ran 𝑅)𝑏)
2217, 20, 213bitr3i 290 . . . . . . . . 9 (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑎(dom 𝑅 × ran 𝑅)𝑏)
2316breqi 4691 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑐(dom 𝑅 × ran 𝑅)𝑏)
24 vex 3234 . . . . . . . . . . 11 𝑐 ∈ V
2524, 18brcnv 5337 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
2624, 18brcnv 5337 . . . . . . . . . 10 (𝑐(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐)
2723, 25, 263bitr3i 290 . . . . . . . . 9 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐𝑏(dom 𝑅 × ran 𝑅)𝑐)
2822, 27anbi12i 733 . . . . . . . 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 206 . . . . . . 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 1802 . . . . . 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 5032 . . . . 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 5152 . . . . 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 5152 . . . . 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 3677 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35 xptrrel 13765 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36 ssun2 3810 . . . . 5 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3735, 36sstri 3645 . . . 4 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3834, 37sstri 3645 . . 3 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39 trcleq2lem 13776 . . . . 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 3383 . . . 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 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 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}))
422, 38, 41mp2ani 714 . 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 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  Vcvv 3231  cun 3605  wss 3607   class class class wbr 4685  {copab 4745   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  ccom 5147  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155
This theorem is referenced by:  trclubi  13781  trclubgi  13782  trclub  13783  trclubg  13784
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