Theorem intop 44130

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 44126	. . 3 ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑚 × 𝑚)))
2	1	elmpocl 7387	. 2 ⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → (𝑀 ∈ V ∧ 𝑁 ∈ V))
3		intopval 44129	. . . 4 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))
4	3	eleq2d 2898	. . 3 ⊢ ((𝑀 ∈ V ∧ 𝑁 ∈ V) → ( ⚬ ∈ (𝑀 intOp 𝑁) ↔ ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀))))
5		elmapi 8428	. . 3 ⊢ ( ⚬ ∈ (𝑁 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
6	4, 5	syl6bi 255	. 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 398   ∈ wcel 2114  Vcvv 3494   × cxp 5553  ⟶wf 6351  (class class class)co 7156   ↑_m cmap 8406   intOp cintop 44123

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-map 8408  df-intop 44126

This theorem is referenced by: (None)