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Theorem trclfv 13948
 Description: The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
Assertion
Ref Expression
trclfv (𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
Distinct variable group:   𝑥,𝑅
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem trclfv
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elex 3361 . . 3 (𝑅𝑉𝑅 ∈ V)
2 trclexlem 13942 . . . 4 (𝑅 ∈ V → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)
3 trclubg 13947 . . . 4 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
42, 3ssexd 4936 . . 3 (𝑅 ∈ V → {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V)
51, 4jccir 505 . 2 (𝑅𝑉 → (𝑅 ∈ V ∧ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V))
6 trcleq1 13937 . . 3 (𝑟 = 𝑅 {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
7 df-trcl 13935 . . 3 t+ = (𝑟 ∈ V ↦ {𝑥 ∣ (𝑟𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
86, 7fvmptg 6422 . 2 ((𝑅 ∈ V ∧ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V) → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
95, 8syl 17 1 (𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  {cab 2756  Vcvv 3349   ∪ cun 3719   ⊆ wss 3721  ∩ cint 4609   × cxp 5247  dom cdm 5249  ran crn 5250   ∘ ccom 5253  ‘cfv 6031  t+ctcl 13933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-trcl 13935 This theorem is referenced by:  brtrclfv  13950  trclfvss  13954  trclfvub  13955  trclfvlb  13956  cotrtrclfv  13960  trclun  13962  brtrclfv2  38538
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