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Theorem intopval 46222
Description: The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
intopval ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁m (𝑀 × 𝑀)))

Proof of Theorem intopval
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-intop 46219 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚)))
21a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚))))
3 simpr 486 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑛 = 𝑁)
4 simpl 484 . . . . 5 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑚 = 𝑀)
54sqxpeqd 5666 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
63, 5oveq12d 7376 . . 3 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑛m (𝑚 × 𝑚)) = (𝑁m (𝑀 × 𝑀)))
76adantl 483 . 2 (((𝑀𝑉𝑁𝑊) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑛m (𝑚 × 𝑚)) = (𝑁m (𝑀 × 𝑀)))
8 elex 3462 . . 3 (𝑀𝑉𝑀 ∈ V)
98adantr 482 . 2 ((𝑀𝑉𝑁𝑊) → 𝑀 ∈ V)
10 elex 3462 . . 3 (𝑁𝑊𝑁 ∈ V)
1110adantl 483 . 2 ((𝑀𝑉𝑁𝑊) → 𝑁 ∈ V)
12 ovexd 7393 . 2 ((𝑀𝑉𝑁𝑊) → (𝑁m (𝑀 × 𝑀)) ∈ V)
132, 7, 9, 11, 12ovmpod 7508 1 ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁m (𝑀 × 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3444   × cxp 5632  (class class class)co 7358  cmpo 7360  m cmap 8768   intOp cintop 46216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-intop 46219
This theorem is referenced by:  intop  46223  clintopval  46224
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