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Theorem trclublem 14889
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 14888 . 2 (𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
2 ssun1 4136 . . 3 𝑅 ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3 relcnv 6060 . . . . . . . . . . . . . 14 Rel 𝑅
4 relssdmrn 6224 . . . . . . . . . . . . . 14 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
53, 4ax-mp 5 . . . . . . . . . . . . 13 𝑅 ⊆ (dom 𝑅 × ran 𝑅)
6 ssequn1 4144 . . . . . . . . . . . . 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 6099 . . . . . . . . . . . . 13 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅(dom 𝑅 × ran 𝑅))
9 cnvxp 6113 . . . . . . . . . . . . . . 15 (dom 𝑅 × ran 𝑅) = (ran 𝑅 × dom 𝑅)
10 df-rn 5648 . . . . . . . . . . . . . . . 16 ran 𝑅 = dom 𝑅
11 dfdm4 5855 . . . . . . . . . . . . . . . 16 dom 𝑅 = ran 𝑅
1210, 11xpeq12i 5665 . . . . . . . . . . . . . . 15 (ran 𝑅 × dom 𝑅) = (dom 𝑅 × ran 𝑅)
139, 12eqtri 2761 . . . . . . . . . . . . . 14 (dom 𝑅 × ran 𝑅) = (dom 𝑅 × ran 𝑅)
1413uneq2i 4124 . . . . . . . . . . . . 13 (𝑅(dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
158, 14eqtri 2761 . . . . . . . . . . . 12 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (𝑅 ∪ (dom 𝑅 × ran 𝑅))
167, 15, 133eqtr4i 2771 . . . . . . . . . . 11 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
1716breqi 5115 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑏(dom 𝑅 × ran 𝑅)𝑎)
18 vex 3451 . . . . . . . . . . 11 𝑏 ∈ V
19 vex 3451 . . . . . . . . . . 11 𝑎 ∈ V
2018, 19brcnv 5842 . . . . . . . . . 10 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑎𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏)
2118, 19brcnv 5842 . . . . . . . . . 10 (𝑏(dom 𝑅 × ran 𝑅)𝑎𝑎(dom 𝑅 × ran 𝑅)𝑏)
2217, 20, 213bitr3i 301 . . . . . . . . 9 (𝑎(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑎(dom 𝑅 × ran 𝑅)𝑏)
2316breqi 5115 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑐(dom 𝑅 × ran 𝑅)𝑏)
24 vex 3451 . . . . . . . . . . 11 𝑐 ∈ V
2524, 18brcnv 5842 . . . . . . . . . 10 (𝑐(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑏𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐)
2624, 18brcnv 5842 . . . . . . . . . 10 (𝑐(dom 𝑅 × ran 𝑅)𝑏𝑏(dom 𝑅 × ran 𝑅)𝑐)
2723, 25, 263bitr3i 301 . . . . . . . . 9 (𝑏(𝑅 ∪ (dom 𝑅 × ran 𝑅))𝑐𝑏(dom 𝑅 × ran 𝑅)𝑐)
2822, 27anbi12i 628 . . . . . . . 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 1838 . . . . . 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 5511 . . . . 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 5646 . . . . 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 5646 . . . . 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 3991 . . . 4 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅))
35 xptrrel 14874 . . . . 5 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (dom 𝑅 × ran 𝑅)
36 ssun2 4137 . . . . 5 (dom 𝑅 × ran 𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3735, 36sstri 3957 . . . 4 ((dom 𝑅 × ran 𝑅) ∘ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
3834, 37sstri 3957 . . 3 ((𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
39 trcleq2lem 14885 . . . . 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 3632 . . . 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 248 . . 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 697 . 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 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  Vcvv 3447  cun 3912  wss 3914   class class class wbr 5109  {copab 5171   × cxp 5635  ccnv 5636  dom cdm 5637  ran crn 5638  ccom 5641  Rel wrel 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649
This theorem is referenced by:  trclubi  14890  trclubgi  14891  trclub  14892  trclubg  14893
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