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Theorem rntrcl 38252
 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 13784 . . . 4 (𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑋 ∪ (dom 𝑋 × ran 𝑋)))
2 rnss 5386 . . . 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 5576 . . . 4 ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = (ran 𝑋 ∪ ran (dom 𝑋 × ran 𝑋))
5 rnxpss 5601 . . . . 5 ran (dom 𝑋 × ran 𝑋) ⊆ ran 𝑋
6 ssequn2 3819 . . . . 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 2673 . . 3 ran (𝑋 ∪ (dom 𝑋 × ran 𝑋)) = ran 𝑋
93, 8syl6sseq 3684 . 2 (𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ ran 𝑋)
10 ssmin 4528 . . 3 𝑋 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)}
11 rnss 5386 . . 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 3653 1 (𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = ran 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637   ∪ cun 3605   ⊆ wss 3607  ∩ cint 4507   × cxp 5141  dom cdm 5143  ran crn 5144   ∘ ccom 5147 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155 This theorem is referenced by:  dfrtrcl5  38253
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