Theorem relexp0d 14735

Description: A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)

Hypotheses

Ref	Expression
relexp0d.1	⊢ (𝜑 → Rel 𝑅)
relexp0d.2	⊢ (𝜑 → 𝑅 ∈ 𝑉)

Assertion

Ref	Expression
relexp0d	⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))

Proof of Theorem relexp0d

Step	Hyp	Ref	Expression
1		relexp0d.2	. 2 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
2		relexp0d.1	. 2 ⊢ (𝜑 → Rel 𝑅)
3		relexp0 14734	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∪ cuni 4839   I cid 5488   ↾ cres 5591  Rel wrel 5594  (class class class)co 7275  0cc0 10871  ↑_𝑟crelexp 14730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-mulcl 10933  ax-i2m1 10939

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-n0 12234  df-relexp 14731

This theorem is referenced by:  rtrclreclem2  14770  rtrclreclem4  14772