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Theorem assintopval 44040
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 44036 . 2 assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
2 fveq2 6663 . . 3 (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
3 breq2 5061 . . 3 (𝑚 = 𝑀 → (𝑜 assLaw 𝑚𝑜 assLaw 𝑀))
42, 3rabeqbidv 3483 . 2 (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
5 elex 3510 . 2 (𝑀𝑉𝑀 ∈ V)
6 fvex 6676 . . . 4 ( clIntOp ‘𝑀) ∈ V
76rabex 5226 . . 3 {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V
87a1i 11 . 2 (𝑀𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V)
91, 4, 5, 8fvmptd3 6783 1 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492   class class class wbr 5057  cfv 6348   assLaw casslaw 44019   clIntOp cclintop 44032   assIntOp cassintop 44033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-assintop 44036
This theorem is referenced by:  assintopmap  44041  isassintop  44045  assintopcllaw  44047
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