Theorem assintop 45743

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

Assertion

Ref	Expression
assintop	⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))

Proof of Theorem assintop

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6857	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2		assintopmap 45740	. . . 4 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
3	2	eleq2d 2822	. . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
4		breq1 5092	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
5	4	elrab 3634	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀))
6		elmapi 8700	. . . . 5 ⊢ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
7	6	anim1i 615	. . . 4 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
8	5, 7	sylbi 216	. . 3 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
9	3, 8	syl6bi 252	. 2 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀)))
10	1, 9	mpcom 38	1 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2105  {crab 3403  Vcvv 3441   class class class wbr 5089   × cxp 5612  ⟶wf 6469  ‘cfv 6473  (class class class)co 7329   ↑_m cmap 8678   assLaw casslaw 45718   assIntOp cassintop 45732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-1st 7891  df-2nd 7892  df-map 8680  df-intop 45733  df-clintop 45734  df-assintop 45735

This theorem is referenced by:  assintopasslaw  45747