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Theorem efmnd 18896
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 3499 . . 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 2793 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎m 𝑎) = 𝐵)
94, 8sylan9eqr 2797 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑏 = 𝐵)
109opeq2d 4885 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 eqidd 2736 . . . . . . . . 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 2793 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
1514opeq2d 4885 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16 simpl 482 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑎 = 𝐴)
17 pweq 4619 . . . . . . . . . . . 12 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
1817sneqd 4643 . . . . . . . . . . 11 (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
1918adantr 480 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
2016, 19xpeq12d 5720 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2120fveq2d 6911 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
22 efmnd.4 . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
2321, 22eqtr4di 2793 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
2423opeq2d 4885 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
2510, 15, 24tpeq123d 4753 . . . . 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 3946 . . . 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 18895 . . . 4 EndoFMnd = (𝑎 ∈ V ↦ (𝑎m 𝑎) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
28 tpex 7765 . . . 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 2787 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908  𝒫 cpw 4605  {csn 4631  {ctp 4635  cop 4637   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865  ndxcnx 17227  Basecbs 17245  +gcplusg 17298  TopSetcts 17304  tcpt 17485  EndoFMndcefmnd 18894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-efmnd 18895
This theorem is referenced by:  efmndbas  18897  efmndtset  18905  efmndplusg  18906  symgvalstruct  19429  symgvalstructOLD  19430
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