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Theorem trclubi 13781
 Description: The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 2-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
Hypotheses
Ref Expression
trclubi.rel Rel 𝑅
trclubi.rex 𝑅 ∈ V
Assertion
Ref Expression
trclubi {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)
Distinct variable group:   𝑅,𝑠

Proof of Theorem trclubi
StepHypRef Expression
1 trclubi.rel . . . 4 Rel 𝑅
2 relssdmrn 5694 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3 ssequn1 3816 . . . . 5 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
42, 3sylib 208 . . . 4 (Rel 𝑅 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
51, 4ax-mp 5 . . 3 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
6 trclubi.rex . . . 4 𝑅 ∈ V
7 trclublem 13780 . . . 4 (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)})
86, 7ax-mp 5 . . 3 (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
95, 8eqeltrri 2727 . 2 (dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
10 intss1 4524 . 2 ((dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} → {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅))
119, 10ax-mp 5 1 {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  ∩ cint 4507   × cxp 5141  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-int 4508  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: (None)
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