Theorem assintop 48832

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 6903	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2		assintopmap 48829	. . . 4 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
3	2	eleq2d 2849	. . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
4		breq1 5104	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
5	4	elrab 3651	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀))
6		elmapi 8831	. . . . 5 ⊢ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
7	6	anim1i 624	. . . 4 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
8	5, 7	sylbi 219	. . 3 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
9	3, 8	biimtrdi 255	. 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 399   ∈ wcel 2143  {crab 3415  Vcvv 3455   class class class wbr 5101   × cxp 5646  ⟶wf 6518  ‘cfv 6522  (class class class)co 7397   ↑_m cmap 8809   assLaw casslaw 48807   assIntOp cassintop 48821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-map 8811  df-intop 48822  df-clintop 48823  df-assintop 48824

This theorem is referenced by:  assintopasslaw  48836