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Theorem dmtrcl 42841
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 (𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = dom 𝑋)
Distinct variable group:   𝑥,𝑋
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem dmtrcl
StepHypRef Expression
1 trclubg 14953 . . . 4 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2 dmss 5902 . . . 4 ( {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
31, 2syl 17 . . 3 (𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
4 dmun 5910 . . . 4 dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋))
5 dmxpss 6170 . . . . 5 dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
6 ssequn2 4183 . . . . 5 (dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 ↔ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋)
75, 6mpbi 229 . . . 4 (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋
84, 7eqtri 2759 . . 3 dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = dom 𝑋
93, 8sseqtrdi 4032 . 2 (𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ dom 𝑋)
10 ssmin 4971 . . 3 𝑋 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
11 dmss 5902 . . 3 (𝑋 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} → dom 𝑋 ⊆ dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
1210, 11mp1i 13 . 2 (𝑋𝑉 → dom 𝑋 ⊆ dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
139, 12eqssd 3999 1 (𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = dom 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  {cab 2708  cun 3946  wss 3948   cint 4950   × cxp 5674  dom cdm 5676  ran crn 5677  ccom 5680
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688
This theorem is referenced by:  dfrtrcl5  42843
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