Theorem trclubNEW 44143

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 44142	. 2 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
3		trclubNEW.rel	. . . 4 ⊢ (𝜑 → Rel 𝑅)
4		relssdmrn 6245	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
6		ssequn1 4133	. . 3 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) ↔ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
7	5, 6	sylib 220	. 2 ⊢ (𝜑 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) = (dom 𝑅 × ran 𝑅))
8	2, 7	sseqtrd 3967	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  {cab 2734  Vcvv 3448   ∪ cun 3897   ⊆ wss 3899  ∩ cint 4899   × cxp 5638  dom cdm 5640  ran crn 5641   ∘ ccom 5644  Rel wrel 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-11 2185  ax-ext 2728  ax-sep 5240  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-br 5095  df-opab 5157  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652

This theorem is referenced by: (None)