Theorem clintopval 48234

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 48230	. 2 ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2		id 22	. . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
3	2, 2	oveq12d 7364	. . 3 ⊢ (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4		intopval 48232	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑉) → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
5	4	anidms 566	. . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
6	3, 5	sylan9eqr 2788	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀 ↑m (𝑀 × 𝑀)))
7		elex 3457	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
8		ovexd 7381	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ↑m (𝑀 × 𝑀)) ∈ V)
9	1, 6, 7, 8	fvmptd2 6937	1 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  Vcvv 3436   × cxp 5614  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750   intOp cintop 48226   clIntOp cclintop 48227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-intop 48229  df-clintop 48230

This theorem is referenced by:  assintopmap  48236  isclintop  48237