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Theorem clintopval 45377
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 45373 . 2 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
32, 2oveq12d 7290 . . 3 (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4 intopval 45375 . . . 4 ((𝑀𝑉𝑀𝑉) → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
54anidms 567 . . 3 (𝑀𝑉 → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
63, 5sylan9eqr 2802 . 2 ((𝑀𝑉𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀m (𝑀 × 𝑀)))
7 elex 3449 . 2 (𝑀𝑉𝑀 ∈ V)
8 ovexd 7307 . 2 (𝑀𝑉 → (𝑀m (𝑀 × 𝑀)) ∈ V)
91, 6, 7, 8fvmptd2 6880 1 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  Vcvv 3431   × cxp 5588  cfv 6432  (class class class)co 7272  m cmap 8607   intOp cintop 45369   clIntOp cclintop 45370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6390  df-fun 6434  df-fv 6440  df-ov 7275  df-oprab 7276  df-mpo 7277  df-intop 45372  df-clintop 45373
This theorem is referenced by:  assintopmap  45379  isclintop  45380
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