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Theorem isclintop 45441
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 45438 . . 3 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
21eleq2d 2819 . 2 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ ∈ (𝑀m (𝑀 × 𝑀))))
3 sqxpexg 7625 . . 3 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 elmapg 8648 . . 3 ((𝑀𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ∈ (𝑀m (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
53, 4mpdan 683 . 2 (𝑀𝑉 → ( ∈ (𝑀m (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
62, 5bitrd 278 1 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2101  Vcvv 3434   × cxp 5589  wf 6443  cfv 6447  (class class class)co 7295  m cmap 8635   clIntOp cclintop 45431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-map 8637  df-intop 45433  df-clintop 45434
This theorem is referenced by:  clintop  45442  isassintop  45444
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