Theorem bj-coi1 38203

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 38187	. 2 ⊢ (Rel 𝐴 ↔ (𝐴 ↾ V) = 𝐴)
2		coi1res 38199	. . 3 ⊢ (𝐴 ∘ I ) = (𝐴 ↾ V)
3	2	eqeq1i 2768	. 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 1570  Vcvv 3454   I cid 5552   ↾ cres 5660   ∘ ccom 5662  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-co 5667  df-res 5670

This theorem is used by: (None)