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Theorem clintopval 44479
 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 44475 . 2 clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2 id 22 . . . 4 (𝑚 = 𝑀𝑚 = 𝑀)
32, 2oveq12d 7153 . . 3 (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4 intopval 44477 . . . 4 ((𝑀𝑉𝑀𝑉) → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
54anidms 570 . . 3 (𝑀𝑉 → (𝑀 intOp 𝑀) = (𝑀m (𝑀 × 𝑀)))
63, 5sylan9eqr 2855 . 2 ((𝑀𝑉𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀m (𝑀 × 𝑀)))
7 elex 3459 . 2 (𝑀𝑉𝑀 ∈ V)
8 ovexd 7170 . 2 (𝑀𝑉 → (𝑀m (𝑀 × 𝑀)) ∈ V)
91, 6, 7, 8fvmptd2 6753 1 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   × cxp 5517  ‘cfv 6324  (class class class)co 7135   ↑m cmap 8391   intOp cintop 44471   clIntOp cclintop 44472 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-intop 44474  df-clintop 44475 This theorem is referenced by:  assintopmap  44481  isclintop  44482
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