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Theorem trclubi 14348
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 6114 . . . . 5 (Rel 𝑅𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3 ssequn1 4154 . . . . 5 (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
42, 3sylib 220 . . . 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 14347 . . . 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 2908 . 2 (dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)}
10 intss1 4882 . 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 398   = wceq 1530  wcel 2107  {cab 2797  Vcvv 3493  cun 3932  wss 3934   cint 4867   × cxp 5546  dom cdm 5548  ran crn 5549  ccom 5552  Rel wrel 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by: (None)
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