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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 Cartesian 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 4115 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | xpcoidgend.1 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) | |
3 | 1, 2 | eqnetrrid 3016 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) |
4 | 3 | xpcogend 14537 | 1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ≠ wne 2940 ∩ cin 3865 ∅c0 4237 × cxp 5549 ∘ ccom 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-co 5560 |
This theorem is referenced by: xptrrel 14543 relexpxpnnidm 40988 |
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