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Theorem dmtrcl 37436
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 13677 . . . 4 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2 dmss 5285 . . . 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 5293 . . . 4 dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋))
5 dmxpss 5526 . . . . 5 dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋
6 ssequn2 3766 . . . . 5 (dom (dom 𝑋 × ran 𝑋) ⊆ dom 𝑋 ↔ (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋)
75, 6mpbi 220 . . . 4 (dom 𝑋 ∪ dom (dom 𝑋 × ran 𝑋)) = dom 𝑋
84, 7eqtri 2643 . . 3 dom (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = dom 𝑋
93, 8syl6sseq 3632 . 2 (𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ dom 𝑋)
10 ssmin 4463 . . 3 𝑋 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
11 dmss 5285 . . 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 3601 1 (𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = dom 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  cun 3554  wss 3556   cint 4442   × cxp 5074  dom cdm 5076  ran crn 5077  ccom 5080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-opab 4676  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088
This theorem is referenced by:  dfrtrcl5  37438
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