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Theorem assintop 44456
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 6682 . 2 ( ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2 assintopmap 44453 . . . 4 (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
32eleq2d 2878 . . 3 (𝑀 ∈ V → ( ∈ ( assIntOp ‘𝑀) ↔ ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
4 breq1 5036 . . . . 5 (𝑜 = → (𝑜 assLaw 𝑀 assLaw 𝑀))
54elrab 3631 . . . 4 ( ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀))
6 elmapi 8415 . . . . 5 ( ∈ (𝑀m (𝑀 × 𝑀)) → :(𝑀 × 𝑀)⟶𝑀)
76anim1i 617 . . . 4 (( ∈ (𝑀m (𝑀 × 𝑀)) ∧ assLaw 𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
85, 7sylbi 220 . . 3 ( ∈ {𝑜 ∈ (𝑀m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
93, 8syl6bi 256 . 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 399  wcel 2112  {crab 3113  Vcvv 3444   class class class wbr 5033   × cxp 5521  wf 6324  cfv 6328  (class class class)co 7139  m cmap 8393   assLaw casslaw 44431   assIntOp cassintop 44445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-intop 44446  df-clintop 44447  df-assintop 44448
This theorem is referenced by:  assintopasslaw  44460
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