Theorem coxp 49631

Description: Composition with a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
coxp	⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))

Proof of Theorem coxp

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

Step	Hyp	Ref	Expression
1		relco 6110	. 2 ⊢ Rel (𝐴 ∘ (𝐵 × 𝐶))
2		relxp 5679	. 2 ⊢ Rel (𝐵 × (𝐴 “ 𝐶))
3		brxp 5710	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐶)𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	3	anbi1i 635	. . . . 5 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
5		anass 473	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
6	4, 5	bitri 278	. . . 4 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
7	6	exbii 1878	. . 3 ⊢ (∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
8		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
9		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
10	8, 9	opelco 5857	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
11	9	elima2 6068	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦))
12	11	anbi2i 634	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
13		opelxp 5697	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)))
14		19.42v 1983	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
15	12, 13, 14	3bitr4i 306	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
16	7, 10, 15	3bitr4i 306	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)))
17	1, 2, 16	eqrelriiv 5776	1 ⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ⟨cop 4595   class class class wbr 5109   × cxp 5659   “ cima 5664   ∘ ccom 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is referenced by:  cosn  49632