Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clintop Structured version   Visualization version   GIF version

Theorem clintop 42362
 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
StepHypRef Expression
1 elfvex 6362 . 2 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
2 isclintop 42361 . . 3 (𝑀 ∈ V → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))
32biimpd 219 . 2 (𝑀 ∈ V → ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀))
41, 3mpcom 38 1 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2144  Vcvv 3349   × cxp 5247  ⟶wf 6027  ‘cfv 6031   clIntOp cclintop 42351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-intop 42353  df-clintop 42354 This theorem is referenced by:  clintopcllaw  42365
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