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Theorem isclintop 42372
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 42369 . . 3 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))
21eleq2d 2826 . 2 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ ∈ (𝑀𝑚 (𝑀 × 𝑀))))
3 sqxpexg 7130 . . 3 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 elmapg 8039 . . 3 ((𝑀𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
53, 4mpdan 705 . 2 (𝑀𝑉 → ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
62, 5bitrd 268 1 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2140  Vcvv 3341   × cxp 5265  wf 6046  cfv 6050  (class class class)co 6815  𝑚 cmap 8026   clIntOp cclintop 42362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-intop 42364  df-clintop 42365
This theorem is referenced by:  clintop  42373  isassintop  42375
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