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Theorem intop 47259
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 47255 . . 3 intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛m (𝑚 × 𝑚)))
21elmpocl 7656 . 2 ( ∈ (𝑀 intOp 𝑁) → (𝑀 ∈ V ∧ 𝑁 ∈ V))
3 intopval 47258 . . . 4 ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 intOp 𝑁) = (𝑁m (𝑀 × 𝑀)))
43eleq2d 2815 . . 3 ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ∈ (𝑀 intOp 𝑁) ↔ ∈ (𝑁m (𝑀 × 𝑀))))
5 elmapi 8861 . . 3 ( ∈ (𝑁m (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑁)
64, 5syl6bi 253 . 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 395  wcel 2099  Vcvv 3470   × cxp 5670  wf 6538  (class class class)co 7414  m cmap 8838   intOp cintop 47252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8840  df-intop 47255
This theorem is referenced by: (None)
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