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Theorem xpcom 4891
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 289 . . . 4 (∃𝑥 𝑥𝐵 → ((𝑎𝐴𝑐𝐶) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶))))
2 ancom 257 . . . . . . . 8 ((𝑎𝐴𝑥𝐵) ↔ (𝑥𝐵𝑎𝐴))
32anbi1i 439 . . . . . . 7 (((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
4 brxp 4402 . . . . . . . 8 (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎𝐴𝑥𝐵))
5 brxp 4402 . . . . . . . 8 (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥𝐵𝑐𝐶))
64, 5anbi12i 441 . . . . . . 7 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)))
7 anandi 532 . . . . . . 7 ((𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
83, 6, 73bitr4i 205 . . . . . 6 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
98exbii 1512 . . . . 5 (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
10 19.41v 1798 . . . . 5 (∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
119, 10bitr2i 178 . . . 4 ((∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐))
121, 11syl6rbb 190 . . 3 (∃𝑥 𝑥𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎𝐴𝑐𝐶)))
1312opabbidv 3850 . 2 (∃𝑥 𝑥𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)})
14 df-co 4381 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)}
15 df-xp 4378 . 2 (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)}
1613, 14, 153eqtr4g 2113 1 (∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409   class class class wbr 3791  {copab 3844   × cxp 4370  ccom 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-co 4381
This theorem is referenced by: (None)
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