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Theorem assintopval 41633
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 41629 . . 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 6088 . . . 4 (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
4 breq2 4581 . . . 4 (𝑚 = 𝑀 → (𝑜 assLaw 𝑚𝑜 assLaw 𝑀))
53, 4rabeqbidv 3167 . . 3 (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
65adantl 480 . 2 ((𝑀𝑉𝑚 = 𝑀) → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
7 elex 3184 . 2 (𝑀𝑉𝑀 ∈ V)
8 fvex 6098 . . . 4 ( clIntOp ‘𝑀) ∈ V
98rabex 4735 . . 3 {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V
109a1i 11 . 2 (𝑀𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V)
112, 6, 7, 10fvmptd 6182 1 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172   class class class wbr 4577  cmpt 4637  cfv 5790   assLaw casslaw 41612   clIntOp cclintop 41625   assIntOp cassintop 41626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-assintop 41629
This theorem is referenced by:  assintopmap  41634  isassintop  41638  assintopcllaw  41640
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