Theorem intopval 48119

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 48116	. . 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 5678	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
6	3, 5	oveq12d 7412	. . 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 3476	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
9	8	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑀 ∈ V)
10		elex 3476	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
11	10	adantl 481	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑁 ∈ V)
12		ovexd 7429	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑁 ↑m (𝑀 × 𝑀)) ∈ V)
13	2, 7, 9, 11, 12	ovmpod 7548	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3455   × cxp 5644  (class class class)co 7394   ∈ cmpo 7396   ↑_m cmap 8803   intOp cintop 48113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-intop 48116

This theorem is referenced by:  intop  48120  clintopval  48121