Theorem relexp0d 14959

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 14958	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∪ cuni 4865   I cid 5526   ↾ cres 5634  Rel wrel 5637  (class class class)co 7368  0cc0 11038  ↑_𝑟crelexp 14954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-mulcl 11100  ax-i2m1 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-n0 12414  df-relexp 14955

This theorem is referenced by:  rtrclreclem2  14994  rtrclreclem4  14996