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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 47155 | . 2 ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)) | |
2 | id 22 | . . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀) | |
3 | 2, 2 | oveq12d 7423 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀)) |
4 | intopval 47157 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑉) → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) | |
5 | 4 | anidms 566 | . . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
6 | 3, 5 | sylan9eqr 2788 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀 ↑m (𝑀 × 𝑀))) |
7 | elex 3487 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
8 | ovexd 7440 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ↑m (𝑀 × 𝑀)) ∈ V) | |
9 | 1, 6, 7, 8 | fvmptd2 7000 | 1 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 × cxp 5667 ‘cfv 6537 (class class class)co 7405 ↑m cmap 8822 intOp cintop 47151 clIntOp cclintop 47152 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-intop 47154 df-clintop 47155 |
This theorem is referenced by: assintopmap 47161 isclintop 47162 |
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