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Theorem clintopval 47159
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 47155 . 2 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
32, 2oveq12d 7423 . . 3 (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4 intopval 47157 . . . 4 ((𝑀𝑉𝑀𝑉) → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
54anidms 566 . . 3 (𝑀𝑉 → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
63, 5sylan9eqr 2788 . 2 ((𝑀𝑉𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀m (𝑀 × 𝑀)))
7 elex 3487 . 2 (𝑀𝑉𝑀 ∈ V)
8 ovexd 7440 . 2 (𝑀𝑉 → (𝑀m (𝑀 × 𝑀)) ∈ V)
91, 6, 7, 8fvmptd2 7000 1 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468   × cxp 5667  cfv 6537  (class class class)co 7405  m cmap 8822   intOp cintop 47151   clIntOp cclintop 47152
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-pr 5420
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-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-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-intop 47154  df-clintop 47155
This theorem is referenced by:  assintopmap  47161  isclintop  47162
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