Theorem coxp 49392

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 6083	. 2 ⊢ Rel (𝐴 ∘ (𝐵 × 𝐶))
2		relxp 5654	. 2 ⊢ Rel (𝐵 × (𝐴 “ 𝐶))
3		brxp 5685	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐶)𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	3	anbi1i 632	. . . . 5 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
5		anass 471	. . . . 5 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
6	4, 5	bitri 277	. . . 4 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
7	6	exbii 1858	. . 3 ⊢ (∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
8		vex 3448	. . . 4 ⊢ 𝑥 ∈ V
9		vex 3448	. . . 4 ⊢ 𝑦 ∈ V
10	8, 9	opelco 5832	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
11	9	elima2 6041	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦))
12	11	anbi2i 631	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
13		opelxp 5672	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)))
14		19.42v 1963	. . . 4 ⊢ (∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
15	12, 13, 14	3bitr4i 305	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
16	7, 10, 15	3bitr4i 305	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)))
17	1, 2, 16	eqrelriiv 5751	1 ⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))

Syntax hints:   ∧ wa 398   = wceq 1550  ∃wex 1789   ∈ wcel 2132  ⟨cop 4578   class class class wbr 5090   × cxp 5634   “ cima 5639   ∘ ccom 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649

This theorem is referenced by:  cosn  49393