Theorem coxp 48705

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 6093	. 2 ⊢ Rel (𝐴 ∘ (𝐵 × 𝐶))
2		relxp 5670	. 2 ⊢ Rel (𝐵 × (𝐴 “ 𝐶))
3		brxp 5701	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐶)𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	3	anbi1i 624	. . . . 5 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
5		anass 468	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
6	4, 5	bitri 275	. . . 4 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
7	6	exbii 1847	. . 3 ⊢ (∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
8		vex 3461	. . . 4 ⊢ 𝑥 ∈ V
9		vex 3461	. . . 4 ⊢ 𝑦 ∈ V
10	8, 9	opelco 5849	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
11	9	elima2 6051	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦))
12	11	anbi2i 623	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
13		opelxp 5688	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)))
14		19.42v 1952	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
15	12, 13, 14	3bitr4i 303	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
16	7, 10, 15	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)))
17	1, 2, 16	eqrelriiv 5767	1 ⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ⟨cop 4605   class class class wbr 5117   × cxp 5650   “ cima 5655   ∘ ccom 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5118  df-opab 5180  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665

This theorem is referenced by:  cosn  48706