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Theorem xpcom 5125
 Description: Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
Assertion
Ref Expression
xpcom (∃𝑥 𝑥𝐵 → ((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.
StepHypRef Expression
1 ibar 299 . . . 4 (∃𝑥 𝑥𝐵 → ((𝑎𝐴𝑐𝐶) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶))))
2 ancom 264 . . . . . . . 8 ((𝑎𝐴𝑥𝐵) ↔ (𝑥𝐵𝑎𝐴))
32anbi1i 454 . . . . . . 7 (((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
4 brxp 4610 . . . . . . . 8 (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎𝐴𝑥𝐵))
5 brxp 4610 . . . . . . . 8 (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥𝐵𝑐𝐶))
64, 5anbi12i 456 . . . . . . 7 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)))
7 anandi 580 . . . . . . 7 ((𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
83, 6, 73bitr4i 211 . . . . . 6 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
98exbii 1582 . . . . 5 (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
10 19.41v 1879 . . . . 5 (∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
119, 10bitr2i 184 . . . 4 ((∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐))
121, 11bitr2di 196 . . 3 (∃𝑥 𝑥𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎𝐴𝑐𝐶)))
1312opabbidv 4026 . 2 (∃𝑥 𝑥𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)})
14 df-co 4588 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)}
15 df-xp 4585 . 2 (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)}
1613, 14, 153eqtr4g 2212 1 (∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 2125   class class class wbr 3961  {copab 4020   × cxp 4577   ∘ ccom 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-xp 4585  df-co 4588 This theorem is referenced by: (None)
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