Theorem trclubi 15175

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 6286	. . . . 5 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
3		ssequn1 4187	. . . . 5 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
4	2, 3	sylib 221	. . . 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 15174	. . . 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 2861	. 2 ⊢ (dom 𝑅 × ran 𝑅) ∈ {𝑠 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)}
10		intss1 4926	. 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)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∩ cint 4910   × cxp 5653  dom cdm 5655  ran crn 5656   ∘ ccom 5659  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by: (None)