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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 5561. (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 264 | . . 3 ⊢ (𝑢 = 〈𝑥, 𝑦〉 → (𝑧 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶)) | |
2 | 1 | dfoprab4 7753 | . 2 ⊢ {〈𝑢, 𝑧〉 ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)} |
3 | df-xp 5561 | . 2 ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈𝑢, 𝑧〉 ∣ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶)} | |
4 | df-3an 1085 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)) | |
5 | 4 | oprabbii 7221 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶)} |
6 | 2, 3, 5 | 3eqtr4i 2854 | 1 ⊢ ((𝐴 × 𝐵) × 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4573 {copab 5128 × cxp 5553 {coprab 7157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-oprab 7160 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: (None) |
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