Theorem dmtrcl 43983

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

Assertion

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

Distinct variable group:   𝑥,𝑋

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem dmtrcl

Step	Hyp	Ref	Expression
1		trclubg 14934	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2		dmss 5859	. . . 4 ⊢ (∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
3	1, 2	syl 17	. . 3 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
4		dmun 5867	. . . 4 ⊢ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋))
5		dmxpss 6137	. . . . 5 ⊢ dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
6		ssequn2 4143	. . . . 5 ⊢ (dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 ↔ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋)
7	5, 6	mpbi 230	. . . 4 ⊢ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋
8	4, 7	eqtri 2760	. . 3 ⊢ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = dom 𝑋
9	3, 8	sseqtrdi 3976	. 2 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom 𝑋)
10		ssmin 4924	. . 3 ⊢ 𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
11		dmss 5859	. . 3 ⊢ (𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} → dom 𝑋 ⊆ dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
12	10, 11	mp1i 13	. 2 ⊢ (𝑋 ∈ 𝑉 → dom 𝑋 ⊆ dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)})
13	9, 12	eqssd 3953	1 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ∪ cun 3901   ⊆ wss 3903  ∩ cint 4904   × cxp 5630  dom cdm 5632  ran crn 5633   ∘ ccom 5636

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

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-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644

This theorem is referenced by:  dfrtrcl5  43985