Theorem assintopval 48160

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

Assertion

Ref	Expression
assintopval	⊢ (𝑀 ∈ 𝑉 → ( 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.

Step	Hyp	Ref	Expression
1		df-assintop 48156	. 2 ⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
2		fveq2 6881	. . 3 ⊢ (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
3		breq2 5128	. . 3 ⊢ (𝑚 = 𝑀 → (𝑜 assLaw 𝑚 ↔ 𝑜 assLaw 𝑀))
4	2, 3	rabeqbidv 3439	. 2 ⊢ (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
5		elex 3485	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
6		fvex 6894	. . . 4 ⊢ ( clIntOp ‘𝑀) ∈ V
7	6	rabex 5314	. . 3 ⊢ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V
8	7	a1i 11	. 2 ⊢ (𝑀 ∈ 𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V)
9	1, 4, 5, 8	fvmptd3 7014	1 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464   class class class wbr 5124  ‘cfv 6536   assLaw casslaw 48139   clIntOp cclintop 48152   assIntOp cassintop 48153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-assintop 48156

This theorem is referenced by:  assintopmap  48161  isassintop  48165  assintopcllaw  48167