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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 3805 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | xpcoidgend.1 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) | |
3 | 1, 2 | syl5eqner 2869 | . 2 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ≠ ∅) |
4 | 3 | xpcogend 13713 | 1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ≠ wne 2794 ∩ cin 3573 ∅c0 3915 × cxp 5112 ∘ ccom 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-co 5123 |
This theorem is referenced by: xptrrel 13719 relexpxpnnidm 37995 |
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