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Theorem clintop 47139
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 6922 . 2 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
2 isclintop 47138 . . 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 2098  Vcvv 3468   × cxp 5667  wf 6532  cfv 6536   clIntOp cclintop 47128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-intop 47130  df-clintop 47131
This theorem is referenced by:  clintopcllaw  47142
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