Theorem coresrn 38197

Description: When composing classes, restricting the second class to the range of the first has no effect. (Contributed by BJ, 22-Oct-2026.) Note: can be moved before cores 6267 and shorten funresfunco 6607, dprdf1 20291, kgencn2 23920, domss2 9191.

Assertion

Ref	Expression
coresrn	⊢ ((𝐵 ↾ ran 𝐴) ∘ 𝐴) = (𝐵 ∘ 𝐴)

Proof of Theorem coresrn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	brelrn 5934	. . . . . 6 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
4		vex 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
5	4	brresi 5993	. . . . . . 7 ⊢ (𝑦(𝐵 ↾ ran 𝐴)𝑧 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑦𝐵𝑧))
6	5	baib 545	. . . . . 6 ⊢ (𝑦 ∈ ran 𝐴 → (𝑦(𝐵 ↾ ran 𝐴)𝑧 ↔ 𝑦𝐵𝑧))
7	3, 6	syl 18	. . . . 5 ⊢ (𝑥𝐴𝑦 → (𝑦(𝐵 ↾ ran 𝐴)𝑧 ↔ 𝑦𝐵𝑧))
8	7	pm5.32i 585	. . . 4 ⊢ ((𝑥𝐴𝑦 ∧ 𝑦(𝐵 ↾ ran 𝐴)𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑦𝐵𝑧))
9	8	exbii 1881	. . 3 ⊢ (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦(𝐵 ↾ ran 𝐴)𝑧) ↔ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐵𝑧))
10	9	opabbii 5176	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦(𝐵 ↾ ran 𝐴)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐵𝑧)}
11		df-co 5665	. 2 ⊢ ((𝐵 ↾ ran 𝐴) ∘ 𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦(𝐵 ↾ ran 𝐴)𝑧)}
12		df-co 5665	. 2 ⊢ (𝐵 ∘ 𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐵𝑧)}
13	10, 11, 12	3eqtr4i 2796	1 ⊢ ((𝐵 ↾ ran 𝐴) ∘ 𝐴) = (𝐵 ∘ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   class class class wbr 5106  {copab 5171  ran crn 5657   ↾ cres 5658   ∘ ccom 5660

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 5253  ax-pr 5396

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 5172  df-xp 5662  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668

This theorem is used by:  coss1b  38209  cossidcnvss2  38215  coidmid  38218  nvo3  38223