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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 46220 | . 2 ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)) | |
2 | id 22 | . . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀) | |
3 | 2, 2 | oveq12d 7376 | . . 3 ⊢ (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀)) |
4 | intopval 46222 | . . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑉) → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) | |
5 | 4 | anidms 568 | . . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
6 | 3, 5 | sylan9eqr 2795 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀 ↑m (𝑀 × 𝑀))) |
7 | elex 3462 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
8 | ovexd 7393 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ↑m (𝑀 × 𝑀)) ∈ V) | |
9 | 1, 6, 7, 8 | fvmptd2 6957 | 1 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 × cxp 5632 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8768 intOp cintop 46216 clIntOp cclintop 46217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-intop 46219 df-clintop 46220 |
This theorem is referenced by: assintopmap 46226 isclintop 46227 |
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