Theorem isclintop 45441

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 45438	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
2	1	eleq2d 2819	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀))))
3		sqxpexg 7625	. . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		elmapg 8648	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
5	3, 4	mpdan 683	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
6	2, 5	bitrd 278	1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2101  Vcvv 3434   × cxp 5589  ⟶wf 6443  ‘cfv 6447  (class class class)co 7295   ↑_m cmap 8635   clIntOp cclintop 45431

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 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-map 8637  df-intop 45433  df-clintop 45434

This theorem is referenced by:  clintop  45442  isassintop  45444