Theorem clintopval 41630

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

Assertion

Ref	Expression
clintopval	⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))

Proof of Theorem clintopval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clintop 41626	. . 3 ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2	1	a1i 11	. 2 ⊢ (𝑀 ∈ 𝑉 → clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)))
3		id 22	. . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
4	3, 3	oveq12d 6567	. . 3 ⊢ (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
5		intopval 41628	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑉) → (𝑀 intOp 𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))
6	5	anidms 675	. . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 intOp 𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))
7	4, 6	sylan9eqr 2666	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))
8		elex 3185	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
9		ovex 6577	. . 3 ⊢ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∈ V
10	9	a1i 11	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ↑𝑚 (𝑀 × 𝑀)) ∈ V)
11	2, 7, 8, 10	fvmptd 6197	1 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   intOp cintop 41622   clIntOp cclintop 41623

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

This theorem is referenced by:  assintopmap  41632  isclintop  41633