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Theorem intop 42367
Description: An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
intop ( ∈ (𝑀 intOp 𝑁) → :(𝑀 × 𝑀)⟶𝑁)

Proof of Theorem intop
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-intop 42363 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚)))
21elmpt2cl 7042 . 2 ( ∈ (𝑀 intOp 𝑁) → (𝑀 ∈ V ∧ 𝑁 ∈ V))
3 intopval 42366 . . . 4 ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))
43eleq2d 2825 . . 3 ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ∈ (𝑀 intOp 𝑁) ↔ ∈ (𝑁𝑚 (𝑀 × 𝑀))))
5 elmapi 8047 . . 3 ( ∈ (𝑁𝑚 (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑁)
64, 5syl6bi 243 . 2 ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ∈ (𝑀 intOp 𝑁) → :(𝑀 × 𝑀)⟶𝑁))
72, 6mpcom 38 1 ( ∈ (𝑀 intOp 𝑁) → :(𝑀 × 𝑀)⟶𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  Vcvv 3340   × cxp 5264  wf 6045  (class class class)co 6814  𝑚 cmap 8025   intOp cintop 42360
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-8 2141  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-pow 4992  ax-pr 5055  ax-un 7115
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-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-intop 42363
This theorem is referenced by: (None)
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