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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isclintop | Structured version Visualization version GIF version |
Description: The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
isclintop | ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clintopval 44464 | . . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) | |
2 | 1 | eleq2d 2875 | . 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)))) |
3 | sqxpexg 7457 | . . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V) | |
4 | elmapg 8402 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) | |
5 | 3, 4 | mpdan 686 | . 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) |
6 | 2, 5 | bitrd 282 | 1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 Vcvv 3441 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 clIntOp cclintop 44457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-intop 44459 df-clintop 44460 |
This theorem is referenced by: clintop 44468 isassintop 44470 |
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