Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 44110 | . . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀))) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)))) |
3 | sqxpexg 7476 | . . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V) | |
4 | elmapg 8418 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) | |
5 | 3, 4 | mpdan 685 | . 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) |
6 | 2, 5 | bitrd 281 | 1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 Vcvv 3494 × cxp 5552 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 clIntOp cclintop 44103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-intop 44105 df-clintop 44106 |
This theorem is referenced by: clintop 44114 isassintop 44116 |
Copyright terms: Public domain | W3C validator |