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Theorem assintopval 46700
Description: The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
assintopval (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Distinct variable group:   𝑜,𝑀
Allowed substitution hint:   𝑉(𝑜)

Proof of Theorem assintopval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 df-assintop 46696 . 2 assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
2 fveq2 6891 . . 3 (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
3 breq2 5152 . . 3 (𝑚 = 𝑀 → (𝑜 assLaw 𝑚𝑜 assLaw 𝑀))
42, 3rabeqbidv 3449 . 2 (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
5 elex 3492 . 2 (𝑀𝑉𝑀 ∈ V)
6 fvex 6904 . . . 4 ( clIntOp ‘𝑀) ∈ V
76rabex 5332 . . 3 {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V
87a1i 11 . 2 (𝑀𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V)
91, 4, 5, 8fvmptd3 7021 1 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3432  Vcvv 3474   class class class wbr 5148  cfv 6543   assLaw casslaw 46679   clIntOp cclintop 46692   assIntOp cassintop 46693
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-assintop 46696
This theorem is referenced by:  assintopmap  46701  isassintop  46705  assintopcllaw  46707
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