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Theorem intopval 42348
 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 42345 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚)))
21a1i 11 . 2 ((𝑀𝑉𝑁𝑊) → intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚))))
3 simpr 479 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑛 = 𝑁)
4 simpl 474 . . . . 5 ((𝑚 = 𝑀𝑛 = 𝑁) → 𝑚 = 𝑀)
54sqxpeqd 5298 . . . 4 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑚 × 𝑚) = (𝑀 × 𝑀))
63, 5oveq12d 6831 . . 3 ((𝑚 = 𝑀𝑛 = 𝑁) → (𝑛𝑚 (𝑚 × 𝑚)) = (𝑁𝑚 (𝑀 × 𝑀)))
76adantl 473 . 2 (((𝑀𝑉𝑁𝑊) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑛𝑚 (𝑚 × 𝑚)) = (𝑁𝑚 (𝑀 × 𝑀)))
8 elex 3352 . . 3 (𝑀𝑉𝑀 ∈ V)
98adantr 472 . 2 ((𝑀𝑉𝑁𝑊) → 𝑀 ∈ V)
10 elex 3352 . . 3 (𝑁𝑊𝑁 ∈ V)
1110adantl 473 . 2 ((𝑀𝑉𝑁𝑊) → 𝑁 ∈ V)
12 ovexd 6843 . 2 ((𝑀𝑉𝑁𝑊) → (𝑁𝑚 (𝑀 × 𝑀)) ∈ V)
132, 7, 9, 11, 12ovmpt2d 6953 1 ((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   × cxp 5264  (class class class)co 6813   ↦ cmpt2 6815   ↑𝑚 cmap 8023   intOp cintop 42342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-intop 42345 This theorem is referenced by:  intop  42349  clintopval  42350
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