Theorem isclintop 41843

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 41840	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑𝑚 (𝑀 × 𝑀)))
2	1	eleq2d 2687	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀))))
3		sqxpexg 6963	. . 3 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
4		elmapg 7870	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝑀 × 𝑀) ∈ V) → ( ⚬ ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
5	3, 4	mpdan 702	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ (𝑀 ↑𝑚 (𝑀 × 𝑀)) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
6	2, 5	bitrd 268	1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990  Vcvv 3200   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   clIntOp cclintop 41833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-intop 41835  df-clintop 41836

This theorem is referenced by:  clintop  41844  isassintop  41846