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Theorem clintopval 46224
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 46220 . 2 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
32, 2oveq12d 7376 . . 3 (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4 intopval 46222 . . . 4 ((𝑀𝑉𝑀𝑉) → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
54anidms 568 . . 3 (𝑀𝑉 → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
63, 5sylan9eqr 2795 . 2 ((𝑀𝑉𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀m (𝑀 × 𝑀)))
7 elex 3462 . 2 (𝑀𝑉𝑀 ∈ V)
8 ovexd 7393 . 2 (𝑀𝑉 → (𝑀m (𝑀 × 𝑀)) ∈ V)
91, 6, 7, 8fvmptd2 6957 1 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3444   × cxp 5632  cfv 6497  (class class class)co 7358  m cmap 8768   intOp cintop 46216   clIntOp cclintop 46217
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 5257  ax-nul 5264  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-intop 46219  df-clintop 46220
This theorem is referenced by:  assintopmap  46226  isclintop  46227
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