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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clintopval | Structured version Visualization version GIF version |
Description: The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
clintopval | ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clintop 47373 | . 2 ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)) | |
2 | id 22 | . . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀) | |
3 | 2, 2 | oveq12d 7433 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀)) |
4 | intopval 47375 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑉) → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) | |
5 | 4 | anidms 565 | . . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
6 | 3, 5 | sylan9eqr 2787 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀 ↑m (𝑀 × 𝑀))) |
7 | elex 3482 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
8 | ovexd 7450 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ↑m (𝑀 × 𝑀)) ∈ V) | |
9 | 1, 6, 7, 8 | fvmptd2 7007 | 1 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 × cxp 5670 ‘cfv 6542 (class class class)co 7415 ↑m cmap 8841 intOp cintop 47369 clIntOp cclintop 47370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-intop 47372 df-clintop 47373 |
This theorem is referenced by: assintopmap 47379 isclintop 47380 |
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