Theorem clintop 47321

Description: A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)

Assertion

Ref	Expression
clintop	⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)

Proof of Theorem clintop

Step	Hyp	Ref	Expression
1		elfvex 6938	. 2 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
2		isclintop 47320	. . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
3	2	biimpd 228	. 2 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀))
4	1, 3	mpcom 38	1 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3471   × cxp 5678  ⟶wf 6547  ‘cfv 6551   clIntOp cclintop 47310

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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851  df-intop 47312  df-clintop 47313

This theorem is referenced by:  clintopcllaw  47324