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Theorem xpcom 4964
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 295 . . . 4 (∃𝑥 𝑥𝐵 → ((𝑎𝐴𝑐𝐶) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶))))
2 ancom 262 . . . . . . . 8 ((𝑎𝐴𝑥𝐵) ↔ (𝑥𝐵𝑎𝐴))
32anbi1i 446 . . . . . . 7 (((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
4 brxp 4458 . . . . . . . 8 (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎𝐴𝑥𝐵))
5 brxp 4458 . . . . . . . 8 (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥𝐵𝑐𝐶))
64, 5anbi12i 448 . . . . . . 7 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)))
7 anandi 557 . . . . . . 7 ((𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
83, 6, 73bitr4i 210 . . . . . 6 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
98exbii 1541 . . . . 5 (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
10 19.41v 1830 . . . . 5 (∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
119, 10bitr2i 183 . . . 4 ((∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐))
121, 11syl6rbb 195 . . 3 (∃𝑥 𝑥𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎𝐴𝑐𝐶)))
1312opabbidv 3896 . 2 (∃𝑥 𝑥𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)})
14 df-co 4437 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)}
15 df-xp 4434 . 2 (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)}
1613, 14, 153eqtr4g 2145 1 (∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wex 1426  wcel 1438   class class class wbr 3837  {copab 3890   × cxp 4426  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-co 4437
This theorem is referenced by: (None)
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