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Theorem assintopval 46225
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 46221 . 2 assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
2 fveq2 6843 . . 3 (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
3 breq2 5110 . . 3 (𝑚 = 𝑀 → (𝑜 assLaw 𝑚𝑜 assLaw 𝑀))
42, 3rabeqbidv 3423 . 2 (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
5 elex 3462 . 2 (𝑀𝑉𝑀 ∈ V)
6 fvex 6856 . . . 4 ( clIntOp ‘𝑀) ∈ V
76rabex 5290 . . 3 {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V
87a1i 11 . 2 (𝑀𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V)
91, 4, 5, 8fvmptd3 6972 1 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3444   class class class wbr 5106  cfv 6497   assLaw casslaw 46204   clIntOp cclintop 46217   assIntOp cassintop 46218
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-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-assintop 46221
This theorem is referenced by:  assintopmap  46226  isassintop  46230  assintopcllaw  46232
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