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Theorem clintopval 47377
Description: The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
clintopval (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))

Proof of Theorem clintopval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 df-clintop 47373 . 2 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
32, 2oveq12d 7433 . . 3 (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4 intopval 47375 . . . 4 ((𝑀𝑉𝑀𝑉) → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
54anidms 565 . . 3 (𝑀𝑉 → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
63, 5sylan9eqr 2787 . 2 ((𝑀𝑉𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀m (𝑀 × 𝑀)))
7 elex 3482 . 2 (𝑀𝑉𝑀 ∈ V)
8 ovexd 7450 . 2 (𝑀𝑉 → (𝑀m (𝑀 × 𝑀)) ∈ V)
91, 6, 7, 8fvmptd2 7007 1 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3463   × cxp 5670  cfv 6542  (class class class)co 7415  m cmap 8841   intOp cintop 47369   clIntOp cclintop 47370
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-intop 47372  df-clintop 47373
This theorem is referenced by:  assintopmap  47379  isclintop  47380
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