Theorem bj-coi1 38067

Description: The identity relation is a right-unit for composition. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) Strengthen to biconditional form and shorten (Revised by BJ, 16-Oct-2026.)

Assertion

Ref	Expression
bj-coi1	⊢ (Rel 𝐴 ↔ (𝐴 ∘ I ) = 𝐴)

Proof of Theorem bj-coi1

Step	Hyp	Ref	Expression
1		bj-dfrel3 38059	. 2 ⊢ (Rel 𝐴 ↔ (𝐴 ↾ V) = 𝐴)
2		coi1res 38066	. . 3 ⊢ (𝐴 ∘ I ) = (𝐴 ↾ V)
3	2	eqeq1i 2765	. 2 ⊢ ((𝐴 ∘ I ) = 𝐴 ↔ (𝐴 ↾ V) = 𝐴)
4	1, 3	bitr4i 281	1 ⊢ (Rel 𝐴 ↔ (𝐴 ∘ I ) = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  Vcvv 3450   I cid 5544   ↾ cres 5652   ∘ ccom 5654  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-co 5659  df-res 5662

This theorem is used by: (None)