Theorem intop 48445

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 48441	. . 3 ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
2	1	elmpocl 7599	. 2 ⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → (𝑀 ∈ V ∧ 𝑁 ∈ V))
3		intopval 48444	. . . 4 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))
4	3	eleq2d 2822	. . 3 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ⚬ ∈ (𝑀 intOp 𝑁) ↔ ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀))))
5		elmapi 8786	. . 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 2113  Vcvv 3440   × cxp 5622  ⟶wf 6488  (class class class)co 7358   ↑_m cmap 8763   intOp cintop 48438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-intop 48441

This theorem is referenced by: (None)