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Theorem isclintop 44935
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 44932 . . 3 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
21eleq2d 2818 . 2 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ ∈ (𝑀m (𝑀 × 𝑀))))
3 sqxpexg 7496 . . 3 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 elmapg 8450 . . 3 ((𝑀𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ∈ (𝑀m (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
53, 4mpdan 687 . 2 (𝑀𝑉 → ( ∈ (𝑀m (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
62, 5bitrd 282 1 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2114  Vcvv 3398   × cxp 5523  wf 6335  cfv 6339  (class class class)co 7170  m cmap 8437   clIntOp cclintop 44925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-intop 44927  df-clintop 44928
This theorem is referenced by:  clintop  44936  isassintop  44938
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