Theorem xpco 6015

Description: Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)

Assertion

Ref	Expression
xpco	⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Proof of Theorem xpco

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

Step	Hyp	Ref	Expression
1		n0 4230	. . . . . 6 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
2	1	biimpi 217	. . . . 5 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵)
3	2	biantrurd 533	. . . 4 ⊢ (𝐵 ≠ ∅ → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))))
4		ancom 461	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
5	4	anbi1i 623	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
6		brxp 5489	. . . . . . . 8 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7		brxp 5489	. . . . . . . 8 ⊢ (𝑦(𝐵 × 𝐶)𝑧 ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
8	6, 7	anbi12i 626	. . . . . . 7 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
9		anandi 672	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
10	5, 8, 9	3bitr4i 304	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
11	10	exbii 1829	. . . . 5 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
12		19.41v 1927	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ (∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
13	11, 12	bitr2i 277	. . . 4 ⊢ ((∃𝑦 𝑦 ∈ 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) ↔ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧))
14	3, 13	syl6rbb 289	. . 3 ⊢ (𝐵 ≠ ∅ → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)))
15	14	opabbidv 5028	. 2 ⊢ (𝐵 ≠ ∅ → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)})
16		df-co 5452	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐵 × 𝐶)𝑧)}
17		df-xp 5449	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)}
18	15, 16, 17	3eqtr4g 2856	1 ⊢ (𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ≠ wne 2984  ∅c0 4211   class class class wbr 4962  {copab 5024   × cxp 5441   ∘ ccom 5447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-co 5452

This theorem is referenced by:  xpcoid  6016  ustund  22513  ustneism  22515  cosnop  30119