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Theorem assintop 47349
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.
StepHypRef Expression
1 elfvex 6940 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopmap 47346 . . . 4 (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
32eleq2d 2815 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
4 breq1 5155 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3684 . . . 4 ( ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀))
6 elmapi 8874 . . . . 5 ( ∈ (𝑀m (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76anim1i 613 . . . 4 (( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
85, 7sylbi 216 . . 3 ( ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
93, 8biimtrdi 252 . 2 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀)))
101, 9mpcom 38 1 ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  {crab 3430  Vcvv 3473   class class class wbr 5152   × cxp 5680  wf 6549  cfv 6553  (class class class)co 7426  m cmap 8851   assLaw casslaw 47324   assIntOp cassintop 47338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-intop 47339  df-clintop 47340  df-assintop 47341
This theorem is referenced by:  assintopasslaw  47353
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