Theorem intopval 42366

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 42363	. . 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 471	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
4		simpl 468	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑚 = 𝑀)
5	4	sqxpeqd 5281	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
6	3, 5	oveq12d 6811	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑛 ↑𝑚 (𝑚 × 𝑚)) = (𝑁 ↑𝑚 (𝑀 × 𝑀)))
7	6	adantl 467	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑛 ↑𝑚 (𝑚 × 𝑚)) = (𝑁 ↑𝑚 (𝑀 × 𝑀)))
8		elex 3364	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
9	8	adantr 466	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑀 ∈ V)
10		elex 3364	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
11	10	adantl 467	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑁 ∈ V)
12		ovexd 6825	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑁 ↑𝑚 (𝑀 × 𝑀)) ∈ V)
13	2, 7, 9, 11, 12	ovmpt2d 6935	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑𝑚 (𝑀 × 𝑀)))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351   × cxp 5247  (class class class)co 6793   ↦ cmpt2 6795   ↑_𝑚 cmap 8009   intOp cintop 42360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-intop 42363

This theorem is referenced by:  intop  42367  clintopval  42368