Theorem dmtrcl 38901

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 14151	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2		dmss 5570	. . . 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 5578	. . . 4 ⊢ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋))
5		dmxpss 5821	. . . . 5 ⊢ dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
6		ssequn2 4009	. . . . 5 ⊢ (dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 ↔ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋)
7	5, 6	mpbi 222	. . . 4 ⊢ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋
8	4, 7	eqtri 2802	. . 3 ⊢ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = dom 𝑋
9	3, 8	syl6sseq 3870	. 2 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ dom 𝑋)
10		ssmin 4731	. . 3 ⊢ 𝑋 ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}
11		dmss 5570	. . 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 3838	1 ⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {cab 2763   ∪ cun 3790   ⊆ wss 3792  ∩ cint 4712   × cxp 5355  dom cdm 5357  ran crn 5358   ∘ ccom 5361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369

This theorem is referenced by:  dfrtrcl5  38903