Theorem xpcom 5283

Description: Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)

Assertion

Ref	Expression
xpcom	⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem xpcom

Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ibar 301	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶))))
2		ancom 266	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴))
3	2	anbi1i 458	. . . . . . 7 ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
4		brxp 4756	. . . . . . . 8 ⊢ (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
5		brxp 4756	. . . . . . . 8 ⊢ (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))
6	4, 5	anbi12i 460	. . . . . . 7 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
7		anandi 594	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)))
8	3, 6, 7	3bitr4i 212	. . . . . 6 ⊢ ((𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
9	8	exbii 1653	. . . . 5 ⊢ (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
10		19.41v 1951	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ (∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
11	9, 10	bitr2i 185	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐵 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐))
12	1, 11	bitr2di 197	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)))
13	12	opabbidv 4155	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)})
14		df-co 4734	. 2 ⊢ ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥 ∧ 𝑥(𝐵 × 𝐶)𝑐)}
15		df-xp 4731	. 2 ⊢ (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)}
16	13, 14, 15	3eqtr4g 2289	1 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202   class class class wbr 4088  {copab 4149   × cxp 4723   ∘ ccom 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-co 4734

This theorem is referenced by: (None)