Theorem intop 47126

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 47122	. . 3 ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
2	1	elmpocl 7642	. 2 ⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → (𝑀 ∈ V ∧ 𝑁 ∈ V))
3		intopval 47125	. . . 4 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))
4	3	eleq2d 2811	. . 3 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ⚬ ∈ (𝑀 intOp 𝑁) ↔ ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀))))
5		elmapi 8840	. . 3 ⊢ ( ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
6	4, 5	syl6bi 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 2098  Vcvv 3466   × cxp 5665  ⟶wf 6530  (class class class)co 7402   ↑_m cmap 8817   intOp cintop 47119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819  df-intop 47122

This theorem is referenced by: (None)