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Theorem isclintop 41628
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 41625 . . 3 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))
21eleq2d 2672 . 2 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ ∈ (𝑀𝑚 (𝑀 × 𝑀))))
3 sqxpexg 6838 . . 3 (𝑀𝑉 → (𝑀 × 𝑀) ∈ V)
4 elmapg 7734 . . 3 ((𝑀𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
53, 4mpdan 698 . 2 (𝑀𝑉 → ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ↔ :(𝑀 × 𝑀)⟶𝑀))
62, 5bitrd 266 1 (𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wcel 1976  Vcvv 3172   × cxp 5026  wf 5786  cfv 5790  (class class class)co 6527  𝑚 cmap 7721   clIntOp cclintop 41618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7723  df-intop 41620  df-clintop 41621
This theorem is referenced by:  clintop  41629  isassintop  41631
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