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Theorem clintop 46618
Description: A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
clintop ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)

Proof of Theorem clintop
StepHypRef Expression
1 elfvex 6930 . 2 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
2 isclintop 46617 . . 3 (𝑀 ∈ V → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
32biimpd 228 . 2 (𝑀 ∈ V → ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀))
41, 3mpcom 38 1 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3475   × cxp 5675  wf 6540  cfv 6544   clIntOp cclintop 46607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-intop 46609  df-clintop 46610
This theorem is referenced by:  clintopcllaw  46621
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