Theorem relexp0d 15037

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 15036	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
4	1, 2, 3	syl2anc 593	1 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ∪ cuni 4865   I cid 5541   ↾ cres 5649  Rel wrel 5652  (class class class)co 7396  0cc0 11073  ↑_𝑟crelexp 15032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-mulcl 11135  ax-i2m1 11141

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-n0 12482  df-relexp 15033

This theorem is referenced by:  rtrclreclem2  15072  rtrclreclem4  15074