ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfxp3 Structured version   GIF version

Theorem dfxp3 5762
Description: Define the cross product of three classes. Compare df-xp 4294. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
dfxp3 ((A × B) × 𝐶) = {⟨⟨x, y⟩, z⟩ ∣ (x A y B z 𝐶)}
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z

Proof of Theorem dfxp3
Dummy variable u is distinct from all other variables.
StepHypRef Expression
1 biidd 161 . . 3 (u = ⟨x, y⟩ → (z 𝐶z 𝐶))
21dfoprab4 5760 . 2 {⟨u, z⟩ ∣ (u (A × B) z 𝐶)} = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z 𝐶)}
3 df-xp 4294 . 2 ((A × B) × 𝐶) = {⟨u, z⟩ ∣ (u (A × B) z 𝐶)}
4 df-3an 886 . . 3 ((x A y B z 𝐶) ↔ ((x A y B) z 𝐶))
54oprabbii 5502 . 2 {⟨⟨x, y⟩, z⟩ ∣ (x A y B z 𝐶)} = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z 𝐶)}
62, 3, 53eqtr4i 2067 1 ((A × B) × 𝐶) = {⟨⟨x, y⟩, z⟩ ∣ (x A y B z 𝐶)}
Colors of variables: wff set class
Syntax hints:   wa 97   w3a 884   = wceq 1242   wcel 1390  cop 3370  {copab 3808   × cxp 4286  {coprab 5456
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-bnd 1396  ax-4 1397  ax-13 1401  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  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-oprab 5459  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator