Theorem isclintop 47930

Description: The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)

Assertion

Ref	Expression
isclintop	⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))

Proof of Theorem isclintop

Step	Hyp	Ref	Expression
1		clintopval 47927	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
2	1	eleq2d 2830	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀))))
3		sqxpexg 7790	. . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		elmapg 8897	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
5	3, 4	mpdan 686	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
6	2, 5	bitrd 279	1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2108  Vcvv 3488   × cxp 5698  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884   clIntOp cclintop 47920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-intop 47922  df-clintop 47923

This theorem is referenced by:  clintop  47931  isassintop  47933