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Theorem xpcom 5053
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 297 . . . 4 (∃𝑥 𝑥𝐵 → ((𝑎𝐴𝑐𝐶) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶))))
2 ancom 264 . . . . . . . 8 ((𝑎𝐴𝑥𝐵) ↔ (𝑥𝐵𝑎𝐴))
32anbi1i 451 . . . . . . 7 (((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
4 brxp 4538 . . . . . . . 8 (𝑎(𝐴 × 𝐵)𝑥 ↔ (𝑎𝐴𝑥𝐵))
5 brxp 4538 . . . . . . . 8 (𝑥(𝐵 × 𝐶)𝑐 ↔ (𝑥𝐵𝑐𝐶))
64, 5anbi12i 453 . . . . . . 7 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ((𝑎𝐴𝑥𝐵) ∧ (𝑥𝐵𝑐𝐶)))
7 anandi 562 . . . . . . 7 ((𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ((𝑥𝐵𝑎𝐴) ∧ (𝑥𝐵𝑐𝐶)))
83, 6, 73bitr4i 211 . . . . . 6 ((𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
98exbii 1567 . . . . 5 (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ ∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
10 19.41v 1856 . . . . 5 (∃𝑥(𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ (∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)))
119, 10bitr2i 184 . . . 4 ((∃𝑥 𝑥𝐵 ∧ (𝑎𝐴𝑐𝐶)) ↔ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐))
121, 11syl6rbb 196 . . 3 (∃𝑥 𝑥𝐵 → (∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐) ↔ (𝑎𝐴𝑐𝐶)))
1312opabbidv 3962 . 2 (∃𝑥 𝑥𝐵 → {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)} = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)})
14 df-co 4516 . 2 ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = {⟨𝑎, 𝑐⟩ ∣ ∃𝑥(𝑎(𝐴 × 𝐵)𝑥𝑥(𝐵 × 𝐶)𝑐)}
15 df-xp 4513 . 2 (𝐴 × 𝐶) = {⟨𝑎, 𝑐⟩ ∣ (𝑎𝐴𝑐𝐶)}
1613, 14, 153eqtr4g 2173 1 (∃𝑥 𝑥𝐵 → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wex 1451  wcel 1463   class class class wbr 3897  {copab 3956   × cxp 4505  ccom 4511
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-co 4516
This theorem is referenced by: (None)
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