Theorem trclubNEW 43897

Description: If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)

Hypotheses

Ref	Expression
trclubNEW.rex	⊢ (𝜑 → 𝑅 ∈ V)
trclubNEW.rel	⊢ (𝜑 → Rel 𝑅)

Assertion

Ref	Expression
trclubNEW	⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))

Distinct variable group:   𝑥,𝑅

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem trclubNEW

Step	Hyp	Ref	Expression
1		trclubNEW.rex	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2	1	trclubgNEW 43896	. 2 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3		trclubNEW.rel	. . . 4 ⊢ (𝜑 → Rel 𝑅)
4		relssdmrn 6226	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		ssequn1 4137	. . 3 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
7	5, 6	sylib 218	. 2 ⊢ (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
8	2, 7	sseqtrd 3969	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2713  Vcvv 3439   ∪ cun 3898   ⊆ wss 3900  ∩ cint 4901   × cxp 5621  dom cdm 5623  ran crn 5624   ∘ ccom 5627  Rel wrel 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635

This theorem is referenced by: (None)