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Theorem assintopmap 41624
Description: The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
assintopmap (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
Distinct variable group:   𝑜,𝑀
Allowed substitution hint:   𝑉(𝑜)

Proof of Theorem assintopmap
StepHypRef Expression
1 assintopval 41623 . 2 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
2 clintopval 41622 . . 3 (𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))
3 rabeq 3166 . . 3 (( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)) → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
42, 3syl 17 . 2 (𝑀𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
51, 4eqtrd 2644 1 (𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  {crab 2900   class class class wbr 4578   × cxp 5026  cfv 5790  (class class class)co 6527  𝑚 cmap 7722   assLaw casslaw 41602   clIntOp cclintop 41615   assIntOp cassintop 41616
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 4704  ax-nul 4712  ax-pr 4828
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 4368  df-br 4579  df-opab 4639  df-mpt 4640  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-ov 6530  df-oprab 6531  df-mpt2 6532  df-intop 41617  df-clintop 41618  df-assintop 41619
This theorem is referenced by:  assintop  41627  isassintop  41628
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