Theorem relexp0d 14995

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 14994	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∪ cuni 4903   I cid 5569   ↾ cres 5674  Rel wrel 5677  (class class class)co 7414  0cc0 11130  ↑_𝑟crelexp 14990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-mulcl 11192  ax-i2m1 11198

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-n0 12495  df-relexp 14991

This theorem is referenced by:  rtrclreclem2  15030  rtrclreclem4  15032