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Theorem assintopval 41631
 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 41627 . . 3 assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
21a1i 11 . 2 (𝑀𝑉 → assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚}))
3 fveq2 6103 . . . 4 (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
4 breq2 4587 . . . 4 (𝑚 = 𝑀 → (𝑜 assLaw 𝑚𝑜 assLaw 𝑀))
53, 4rabeqbidv 3168 . . 3 (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
65adantl 481 . 2 ((𝑀𝑉𝑚 = 𝑀) → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
7 elex 3185 . 2 (𝑀𝑉𝑀 ∈ V)
8 fvex 6113 . . . 4 ( clIntOp ‘𝑀) ∈ V
98rabex 4740 . . 3 {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V
109a1i 11 . 2 (𝑀𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V)
112, 6, 7, 10fvmptd 6197 1 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804   assLaw casslaw 41610   clIntOp cclintop 41623   assIntOp cassintop 41624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-assintop 41627 This theorem is referenced by:  assintopmap  41632  isassintop  41636  assintopcllaw  41638
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