Theorem coxp 49308

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 6073	. 2 ⊢ Rel (𝐴 ∘ (𝐵 × 𝐶))
2		relxp 5649	. 2 ⊢ Rel (𝐵 × (𝐴 “ 𝐶))
3		brxp 5680	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐶)𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	3	anbi1i 625	. . . . 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 1850	. . 3 ⊢ (∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
8		vex 3433	. . . 4 ⊢ 𝑥 ∈ V
9		vex 3433	. . . 4 ⊢ 𝑦 ∈ V
10	8, 9	opelco 5826	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
11	9	elima2 6031	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦))
12	11	anbi2i 624	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
13		opelxp 5667	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)))
14		19.42v 1955	. . . 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 5746	1 ⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085   × cxp 5629   “ cima 5634   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  cosn  49309