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Theorem assintop 41635
 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 6131 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopmap 41632 . . . 4 (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
32eleq2d 2673 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
4 breq1 4586 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3331 . . . 4 ( ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀))
6 elmapi 7765 . . . . 5 ( ∈ (𝑀𝑚 (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76anim1i 590 . . . 4 (( ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
85, 7sylbi 206 . . 3 ( ∈ {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
93, 8syl6bi 242 . 2 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀)))
101, 9mpcom 37 1 ( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   assLaw casslaw 41610   assIntOp cassintop 41624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-intop 41625  df-clintop 41626  df-assintop 41627 This theorem is referenced by:  assintopasslaw  41639
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