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Theorem trclubi 13936
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 5817 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3 ssequn1 3926 . . . . 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 13935 . . . 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 2836 . 2 (dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
10 intss1 4644 . 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 1632  wcel 2139  {cab 2746  Vcvv 3340  cun 3713  wss 3715   cint 4627   × cxp 5264  dom cdm 5266  ran crn 5267  ccom 5270  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278
This theorem is referenced by: (None)
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