Theorem xpcom 4807

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 285	. . . 4 ⊢ (∃x x ∈ B → ((𝑎 ∈ A ∧ 𝑐 ∈ 𝐶) ↔ (∃x x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶))))
2		ancom 253	. . . . . . . 8 ⊢ ((𝑎 ∈ A ∧ x ∈ B) ↔ (x ∈ B ∧ 𝑎 ∈ A))
3	2	anbi1i 431	. . . . . . 7 ⊢ (((𝑎 ∈ A ∧ x ∈ B) ∧ (x ∈ B ∧ 𝑐 ∈ 𝐶)) ↔ ((x ∈ B ∧ 𝑎 ∈ A) ∧ (x ∈ B ∧ 𝑐 ∈ 𝐶)))
4		brxp 4318	. . . . . . . 8 ⊢ (𝑎(A × B)x ↔ (𝑎 ∈ A ∧ x ∈ B))
5		brxp 4318	. . . . . . . 8 ⊢ (x(B × 𝐶)𝑐 ↔ (x ∈ B ∧ 𝑐 ∈ 𝐶))
6	4, 5	anbi12i 433	. . . . . . 7 ⊢ ((𝑎(A × B)x ∧ x(B × 𝐶)𝑐) ↔ ((𝑎 ∈ A ∧ x ∈ B) ∧ (x ∈ B ∧ 𝑐 ∈ 𝐶)))
7		anandi 524	. . . . . . 7 ⊢ ((x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)) ↔ ((x ∈ B ∧ 𝑎 ∈ A) ∧ (x ∈ B ∧ 𝑐 ∈ 𝐶)))
8	3, 6, 7	3bitr4i 201	. . . . . 6 ⊢ ((𝑎(A × B)x ∧ x(B × 𝐶)𝑐) ↔ (x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)))
9	8	exbii 1493	. . . . 5 ⊢ (∃x(𝑎(A × B)x ∧ x(B × 𝐶)𝑐) ↔ ∃x(x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)))
10		19.41v 1779	. . . . 5 ⊢ (∃x(x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)) ↔ (∃x x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)))
11	9, 10	bitr2i 174	. . . 4 ⊢ ((∃x x ∈ B ∧ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)) ↔ ∃x(𝑎(A × B)x ∧ x(B × 𝐶)𝑐))
12	1, 11	syl6rbb 186	. . 3 ⊢ (∃x x ∈ B → (∃x(𝑎(A × B)x ∧ x(B × 𝐶)𝑐) ↔ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)))
13	12	opabbidv 3814	. 2 ⊢ (∃x x ∈ B → {⟨𝑎, 𝑐⟩ ∣ ∃x(𝑎(A × B)x ∧ x(B × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)})
14		df-co 4297	. 2 ⊢ ((B × 𝐶) ∘ (A × B)) = {⟨𝑎, 𝑐⟩ ∣ ∃x(𝑎(A × B)x ∧ x(B × 𝐶)𝑐)}
15		df-xp 4294	. 2 ⊢ (A × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎 ∈ A ∧ 𝑐 ∈ 𝐶)}
16	13, 14, 15	3eqtr4g 2094	1 ⊢ (∃x x ∈ B → ((B × 𝐶) ∘ (A × B)) = (A × 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390   class class class wbr 3755  {copab 3808   × cxp 4286   ∘ ccom 4292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-co 4297

This theorem is referenced by: (None)