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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 4001 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | xpcoidgend.1 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) | |
3 | 1, 2 | syl5eqner 3044 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) |
4 | 3 | xpcogend 14053 | 1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ≠ wne 2969 ∩ cin 3766 ∅c0 4113 × cxp 5308 ∘ ccom 5314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-opab 4904 df-xp 5316 df-co 5319 |
This theorem is referenced by: xptrrel 14059 relexpxpnnidm 38766 |
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