Theorem coires2 6283

Description: Postcomposing with a restricted identity relation is the same as corestricting. (Contributed by BJ, 1-Nov-2026.)

Assertion

Ref	Expression
coires2	⊢ (( I ↾ 𝐵) ∘ 𝐴) = (𝐴 ∩ (V × 𝐵))

Proof of Theorem coires2

Step	Hyp	Ref	Expression
1		cnvresid 6281	. . 3 ⊢ ◡( I ↾ 𝐵) = ( I ↾ 𝐵)
2	1	coeq1i 5843	. 2 ⊢ (◡( I ↾ 𝐵) ∘ ◡◡𝐴) = (( I ↾ 𝐵) ∘ ◡◡𝐴)
3		coires1 6282	. . . 4 ⊢ (◡𝐴 ∘ ( I ↾ 𝐵)) = (◡𝐴 ↾ 𝐵)
4	3	cnveqi 5859	. . 3 ⊢ ◡(◡𝐴 ∘ ( I ↾ 𝐵)) = ◡(◡𝐴 ↾ 𝐵)
5		cnvco 5874	. . 3 ⊢ ◡(◡𝐴 ∘ ( I ↾ 𝐵)) = (◡( I ↾ 𝐵) ∘ ◡◡𝐴)
6		cnvrescnv 6200	. . 3 ⊢ ◡(◡𝐴 ↾ 𝐵) = (𝐴 ∩ (V × 𝐵))
7	4, 5, 6	3eqtr3i 2794	. 2 ⊢ (◡( I ↾ 𝐵) ∘ ◡◡𝐴) = (𝐴 ∩ (V × 𝐵))
8		cocnvcnv2 6273	. 2 ⊢ (( I ↾ 𝐵) ∘ ◡◡𝐴) = (( I ↾ 𝐵) ∘ 𝐴)
9	2, 7, 8	3eqtr3ri 2795	1 ⊢ (( I ↾ 𝐵) ∘ 𝐴) = (𝐴 ∩ (V × 𝐵))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∩ cin 4092   I cid 5549   × cxp 5653  ^◡ccnv 5654   ↾ cres 5657   ∘ ccom 5659

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 5252  ax-pr 5395

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 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667

This theorem is used by: (None)