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Mirrors > Home > MPE Home > Th. List > xpcoidgend | Structured version Visualization version GIF version |
Description: If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
xpcoidgend.1 | ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) |
Ref | Expression |
---|---|
xpcoidgend | ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4175 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | xpcoidgend.1 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) | |
3 | 1, 2 | eqnetrrid 3088 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) |
4 | 3 | xpcogend 14322 | 1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ≠ wne 3013 ∩ cin 3932 ∅c0 4288 × cxp 5546 ∘ ccom 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-co 5557 |
This theorem is referenced by: xptrrel 14328 relexpxpnnidm 39926 |
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