Theorem intopval 48046

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

Assertion

Ref	Expression
intopval	⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))

Proof of Theorem intopval

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-intop 48043	. . 3 ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
2	1	a1i 11	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚))))
3		simpr 484	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
4		simpl 482	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑚 = 𝑀)
5	4	sqxpeqd 5721	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
6	3, 5	oveq12d 7449	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑛 ↑m (𝑚 × 𝑚)) = (𝑁 ↑m (𝑀 × 𝑀)))
7	6	adantl 481	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑛 ↑m (𝑚 × 𝑚)) = (𝑁 ↑m (𝑀 × 𝑀)))
8		elex 3499	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
9	8	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑀 ∈ V)
10		elex 3499	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
11	10	adantl 481	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑁 ∈ V)
12		ovexd 7466	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑁 ↑m (𝑀 × 𝑀)) ∈ V)
13	2, 7, 9, 11, 12	ovmpod 7585	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   × cxp 5687  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8865   intOp cintop 48040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-intop 48043

This theorem is referenced by:  intop  48047  clintopval  48048