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Theorem rntrcl 37413
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 (𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = ran 𝑋)
Distinct variable group:   𝑥,𝑋
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem rntrcl
StepHypRef Expression
1 trclubg 13674 . . . 4 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2 rnss 5314 . . . 4 ( {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)) → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
31, 2syl 17 . . 3 (𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
4 rnun 5500 . . . 4 ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋))
5 rnxpss 5525 . . . . 5 ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
6 ssequn2 3764 . . . . 5 (ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋 ↔ (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋)) = ran 𝑋)
75, 6mpbi 220 . . . 4 (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋)) = ran 𝑋
84, 7eqtri 2643 . . 3 ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = ran 𝑋
93, 8syl6sseq 3630 . 2 (𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ ran 𝑋)
10 ssmin 4461 . . 3 𝑋 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
11 rnss 5314 . . 3 (𝑋 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} → ran 𝑋 ⊆ ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
1210, 11mp1i 13 . 2 (𝑋𝑉 → ran 𝑋 ⊆ ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
139, 12eqssd 3600 1 (𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = ran 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  cun 3553  wss 3555   cint 4440   × cxp 5072  dom cdm 5074  ran crn 5075  ccom 5078
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086
This theorem is referenced by:  dfrtrcl5  37414
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