Theorem trclubi 14953

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 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

Distinct variable group:   𝑅,𝑠

Proof of Theorem trclubi

Step	Hyp	Ref	Expression
1		trclubi.rel	. . . 4 ⊢ Rel 𝑅
2		relssdmrn 6224	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3		ssequn1 4118	. . . . 5 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
4	2, 3	sylib 220	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
5	1, 4	ax-mp 5	. . 3 ⊢ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅)
6		trclubi.rex	. . . 4 ⊢ 𝑅 ∈ V
7		trclublem 14952	. . . 4 ⊢ (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)})
8	6, 7	ax-mp 5	. . 3 ⊢ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}
9	5, 8	eqeltrri 2838	. 2 ⊢ (dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}
10		intss1 4896	. 2 ⊢ ((dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} → ∩ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅))
11	9, 10	ax-mp 5	1 ⊢ ∩ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

Syntax hints:   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  Vcvv 3433   ∪ cun 3883   ⊆ wss 3885  ∩ cint 4880   × cxp 5619  dom cdm 5621  ran crn 5622   ∘ ccom 5625  Rel wrel 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-ext 2713  ax-sep 5221  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-br 5076  df-opab 5138  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633

This theorem is referenced by: (None)