Theorem assintopmap 41632

Description: The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)

Assertion

Ref	Expression
assintopmap	⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})

Distinct variable group:   𝑜,𝑀

Allowed substitution hint:   𝑉(𝑜)

Proof of Theorem assintopmap

Step	Hyp	Ref	Expression
1		assintopval 41631	. 2 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
2		clintopval 41630	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))
3		rabeq 3166	. . 3 ⊢ (( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)) → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
4	2, 3	syl 17	. 2 ⊢ (𝑀 ∈ 𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
5	1, 4	eqtrd 2644	1 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   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-ov 6552  df-oprab 6553  df-mpt2 6554  df-intop 41625  df-clintop 41626  df-assintop 41627

This theorem is referenced by:  assintop  41635  isassintop  41636