Theorem assintopval 48990

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 48986	. 2 ⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})
2		fveq2 6881	. . 3 ⊢ (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀))
3		breq2 5113	. . 3 ⊢ (𝑚 = 𝑀 → (𝑜 assLaw 𝑚 ↔ 𝑜 assLaw 𝑀))
4	2, 3	rabeqbidv 3434	. 2 ⊢ (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
5		elex 3476	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
6		fvex 6894	. . . 4 ⊢ ( clIntOp ‘𝑀) ∈ V
7	6	rabex 5309	. . 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 7013	1 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   class class class wbr 5109  ‘cfv 6536   assLaw casslaw 48969   clIntOp cclintop 48982   assIntOp cassintop 48983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-assintop 48986

This theorem is referenced by:  assintopmap  48991  isassintop  48995  assintopcllaw  48997