Theorem rntrcl 43811

Description: The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)

Assertion

Ref	Expression
rntrcl	⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)

Distinct variable group:   𝑥,𝑋

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rntrcl

Step	Hyp	Ref	Expression
1		trclubg 14920	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2		rnss 5886	. . . 4 ⊢ (∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
3	1, 2	syl 17	. . 3 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
4		rnun 6101	. . . 4 ⊢ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋))
5		rnxpss 6128	. . . . 5 ⊢ ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
6		ssequn2 4139	. . . . 5 ⊢ (ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋 ↔ (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋)) = ran 𝑋)
7	5, 6	mpbi 230	. . . 4 ⊢ (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋)) = ran 𝑋
8	4, 7	eqtri 2757	. . 3 ⊢ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = ran 𝑋
9	3, 8	sseqtrdi 3972	. 2 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ ran 𝑋)
10		ssmin 4920	. . 3 ⊢ 𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
11		rnss 5886	. . 3 ⊢ (𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} → ran 𝑋 ⊆ ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
12	10, 11	mp1i 13	. 2 ⊢ (𝑋 ∈ 𝑉 → ran 𝑋 ⊆ ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
13	9, 12	eqssd 3949	1 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712   ∪ cun 3897   ⊆ wss 3899  ∩ cint 4900   × cxp 5620  dom cdm 5622  ran crn 5623   ∘ ccom 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634

This theorem is referenced by:  dfrtrcl5  43812