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Mirrors > Home > MPE Home > Th. List > dfxp3 | Structured version Visualization version GIF version |
Description: Define the Cartesian product of three classes. Compare df-xp 5044. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
Ref | Expression |
---|---|
dfxp3 | ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 251 | . . 3 ⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶)) | |
2 | 1 | dfoprab4 7116 | . 2 ⊢ {〈𝑢, 𝑧〉 ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)} |
3 | df-xp 5044 | . 2 ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈𝑢, 𝑧〉 ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} | |
4 | df-3an 1033 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)) | |
5 | 4 | oprabbii 6608 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)} |
6 | 2, 3, 5 | 3eqtr4i 2642 | 1 ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 {copab 4642 × cxp 5036 {coprab 6550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-oprab 6553 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: (None) |
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