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Theorem isclintop 44108
Description: The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
Assertion
Ref Expression
isclintop (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))

Proof of Theorem isclintop
StepHypRef Expression
1 clintopval 44105 . . 3 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
21eleq2d 2898 . 2 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ ∈ (𝑀m (𝑀 × 𝑀))))
3 sqxpexg 7471 . . 3 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 elmapg 8413 . . 3 ((𝑀𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ∈ (𝑀m (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
53, 4mpdan 685 . 2 (𝑀𝑉 → ( ∈ (𝑀m (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
62, 5bitrd 281 1 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wcel 2110  Vcvv 3494   × cxp 5547  wf 6345  cfv 6349  (class class class)co 7150  m cmap 8400   clIntOp cclintop 44098
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 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
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 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-intop 44100  df-clintop 44101
This theorem is referenced by:  clintop  44109  isassintop  44111
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