Theorem rntrcl 43652

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 15018	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2		rnss 5919	. . . 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 6134	. . . 4 ⊢ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋))
5		rnxpss 6161	. . . . 5 ⊢ ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
6		ssequn2 4164	. . . . 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 2758	. . 3 ⊢ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = ran 𝑋
9	3, 8	sseqtrdi 3999	. 2 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ ran 𝑋)
10		ssmin 4943	. . 3 ⊢ 𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
11		rnss 5919	. . 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 3976	1 ⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713   ∪ cun 3924   ⊆ wss 3926  ∩ cint 4922   × cxp 5652  dom cdm 5654  ran crn 5655   ∘ ccom 5658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666

This theorem is referenced by:  dfrtrcl5  43653