Theorem clintopval 49506

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 49502	. 2 ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))
2		id 23	. . . 4 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀)
3	2, 2	oveq12d 7461	. . 3 ⊢ (𝑚 = 𝑀 → (𝑚 intOp 𝑚) = (𝑀 intOp 𝑀))
4		intopval 49504	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑉) → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
5	4	anidms 577	. . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 intOp 𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
6	3, 5	sylan9eqr 2820	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → (𝑚 intOp 𝑚) = (𝑀 ↑m (𝑀 × 𝑀)))
7		elex 3475	. 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
8		ovexd 7478	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ↑m (𝑀 × 𝑀)) ∈ V)
9	1, 6, 7, 8	fvmptd2 7025	1 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   × cxp 5653  ‘cfv 6558  (class class class)co 7443   ↑_m cmap 8875   intOp cintop 49498   clIntOp cclintop 49499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-intop 49501  df-clintop 49502

This theorem is used by:  assintopmap  49508  isclintop  49509