Theorem coxp 49454

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 6097	. 2 ⊢ Rel (𝐴 ∘ (𝐵 × 𝐶))
2		relxp 5665	. 2 ⊢ Rel (𝐵 × (𝐴 “ 𝐶))
3		brxp 5696	. . . . . 6 ⊢ (𝑥(𝐵 × 𝐶)𝑧 ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	3	anbi1i 633	. . . . 5 ⊢ ((𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑧𝐴𝑦))
5		anass 472	. . . . 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 1868	. . 3 ⊢ (∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑥 ∈ 𝐵 ∧ (𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
8		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
9		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
10	8, 9	opelco 5843	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ (𝐵 × 𝐶)) ↔ ∃𝑧(𝑥(𝐵 × 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
11	9	elima2 6055	. . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦))
12	11	anbi2i 632	. . . 4 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧(𝑧 ∈ 𝐶 ∧ 𝑧𝐴𝑦)))
13		opelxp 5683	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × (𝐴 “ 𝐶)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ (𝐴 “ 𝐶)))
14		19.42v 1973	. . . 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 5762	1 ⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))

Syntax hints:   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ⟨cop 4588   class class class wbr 5100   × cxp 5645   “ cima 5650   ∘ ccom 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  cosn  49455