Theorem intop 48047

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.

Step	Hyp	Ref	Expression
1		df-intop 48043	. . 3 ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
2	1	elmpocl 7674	. 2 ⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → (𝑀 ∈ V ∧ 𝑁 ∈ V))
3		intopval 48046	. . . 4 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))
4	3	eleq2d 2825	. . 3 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ⚬ ∈ (𝑀 intOp 𝑁) ↔ ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀))))
5		elmapi 8888	. . 3 ⊢ ( ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
6	4, 5	biimtrdi 253	. 2 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁))
7	2, 6	mpcom 38	1 ⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   × cxp 5687  ⟶wf 6559  (class class class)co 7431   ↑_m cmap 8865   intOp cintop 48040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-intop 48043

This theorem is referenced by: (None)