Theorem rntrcl 41125

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 14638	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2		rnss 5837	. . . 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 6038	. . . 4 ⊢ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋))
5		rnxpss 6064	. . . . 5 ⊢ ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
6		ssequn2 4113	. . . . 5 ⊢ (ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋 ↔ (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋)) = ran 𝑋)
7	5, 6	mpbi 229	. . . 4 ⊢ (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋)) = ran 𝑋
8	4, 7	eqtri 2766	. . 3 ⊢ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = ran 𝑋
9	3, 8	sseqtrdi 3967	. 2 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ ran 𝑋)
10		ssmin 4895	. . 3 ⊢ 𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
11		rnss 5837	. . 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 3934	1 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715   ∪ cun 3881   ⊆ wss 3883  ∩ cint 4876   × cxp 5578  dom cdm 5580  ran crn 5581   ∘ ccom 5584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592

This theorem is referenced by:  dfrtrcl5  41126