Theorem assintopval 48565

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 48561	. 2 ⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
2		fveq2 6842	. . 3 ⊢ (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
3		breq2 5104	. . 3 ⊢ (𝑚 = 𝑀 → (𝑜 assLaw 𝑚 ↔ 𝑜 assLaw 𝑀))
4	2, 3	rabeqbidv 3419	. 2 ⊢ (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
5		elex 3463	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
6		fvex 6855	. . . 4 ⊢ ( clIntOp ‘𝑀) ∈ V
7	6	rabex 5286	. . 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 6973	1 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   class class class wbr 5100  ‘cfv 6500   assLaw casslaw 48544   clIntOp cclintop 48557   assIntOp cassintop 48558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-assintop 48561

This theorem is referenced by:  assintopmap  48566  isassintop  48570  assintopcllaw  48572