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Theorem efmnd 18883
Description: The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
Hypotheses
Ref Expression
efmnd.1 𝐺 = (EndoFMnd‘𝐴)
efmnd.2 𝐵 = (𝐴m 𝐴)
efmnd.3 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
efmnd.4 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
Assertion
Ref Expression
efmnd (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Distinct variable group:   𝑓,𝑔,𝐴
Allowed substitution hints:   𝐵(𝑓,𝑔)   + (𝑓,𝑔)   𝐺(𝑓,𝑔)   𝐽(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem efmnd
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efmnd.1 . 2 𝐺 = (EndoFMnd‘𝐴)
2 elex 3501 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovexd 7466 . . . . 5 (𝑎 = 𝐴 → (𝑎m 𝑎) ∈ V)
4 id 22 . . . . . . . 8 (𝑏 = (𝑎m 𝑎) → 𝑏 = (𝑎m 𝑎))
5 id 22 . . . . . . . . . 10 (𝑎 = 𝐴𝑎 = 𝐴)
65, 5oveq12d 7449 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎m 𝑎) = (𝐴m 𝐴))
7 efmnd.2 . . . . . . . . 9 𝐵 = (𝐴m 𝐴)
86, 7eqtr4di 2795 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎m 𝑎) = 𝐵)
94, 8sylan9eqr 2799 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑏 = 𝐵)
109opeq2d 4880 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 eqidd 2738 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑔) = (𝑓𝑔))
129, 9, 11mpoeq123dv 7508 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
13 efmnd.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
1412, 13eqtr4di 2795 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
1514opeq2d 4880 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16 simpl 482 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑎 = 𝐴)
17 pweq 4614 . . . . . . . . . . . 12 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
1817sneqd 4638 . . . . . . . . . . 11 (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
1918adantr 480 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
2016, 19xpeq12d 5716 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2120fveq2d 6910 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
22 efmnd.4 . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
2321, 22eqtr4di 2795 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
2423opeq2d 4880 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
2510, 15, 24tpeq123d 4748 . . . . 5 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
263, 25csbied 3935 . . . 4 (𝑎 = 𝐴(𝑎m 𝑎) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
27 df-efmnd 18882 . . . 4 EndoFMnd = (𝑎 ∈ V ↦ (𝑎m 𝑎) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
28 tpex 7766 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
2926, 27, 28fvmpt 7016 . . 3 (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
302, 29syl 17 . 2 (𝐴𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
311, 30eqtrid 2789 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  𝒫 cpw 4600  {csn 4626  {ctp 4630  cop 4632   × cxp 5683  ccom 5689  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866  ndxcnx 17230  Basecbs 17247  +gcplusg 17297  TopSetcts 17303  tcpt 17483  EndoFMndcefmnd 18881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-efmnd 18882
This theorem is referenced by:  efmndbas  18884  efmndtset  18892  efmndplusg  18893  symgvalstruct  19414  symgvalstructOLD  19415
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