Theorem dfxp3 5820

Description: Define the cross product of three classes. Compare df-xp 4351. (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.

Step	Hyp	Ref	Expression
1		biidd 161	. . 3 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶))
2	1	dfoprab4 5818	. 2 ⊢ {⟨𝑢, 𝑧⟩ ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)}
3		df-xp 4351	. 2 ⊢ ((𝐴 × 𝐵) × 𝐶) = {⟨𝑢, 𝑧⟩ ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)}
4		df-3an 887	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶))
5	4	oprabbii 5560	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)}
6	2, 3, 5	3eqtr4i 2070	1 ⊢ ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)}

Syntax hints:   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  {copab 3817   × cxp 4343  {coprab 5513

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-oprab 5516  df-1st 5767  df-2nd 5768

This theorem is referenced by: (None)