Theorem intopval 41609

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 41606	. . 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 475	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
4		simpl 471	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → 𝑚 = 𝑀)
5	4	sqxpeqd 5054	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
6	3, 5	oveq12d 6544	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → (𝑛 ↑𝑚 (𝑚 × 𝑚)) = (𝑁 ↑𝑚 (𝑀 × 𝑀)))
7	6	adantl 480	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑛 ↑𝑚 (𝑚 × 𝑚)) = (𝑁 ↑𝑚 (𝑀 × 𝑀)))
8		elex 3184	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
9	8	adantr 479	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑀 ∈ V)
10		elex 3184	. . 3 ⊢ (𝑁 ∈ 𝑊 → 𝑁 ∈ V)
11	10	adantl 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑁 ∈ V)
12		ovex 6554	. . 3 ⊢ (𝑁 ↑𝑚 (𝑀 × 𝑀)) ∈ V
13	12	a1i 11	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑁 ↑𝑚 (𝑀 × 𝑀)) ∈ V)
14	2, 7, 9, 11, 13	ovmpt2d 6663	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑𝑚 (𝑀 × 𝑀)))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1474   ∈ wcel 1976  Vcvv 3172   × cxp 5025  (class class class)co 6526   ↦ cmpt2 6528   ↑_𝑚 cmap 7721   intOp cintop 41603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-intop 41606

This theorem is referenced by:  intop  41610  clintopval  41611