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Theorem efmnd 18672
Description: The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
Hypotheses
Ref Expression
efmnd.1 𝐺 = (EndoFMndβ€˜π΄)
efmnd.2 𝐡 = (𝐴 ↑m 𝐴)
efmnd.3 + = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑓 ∘ 𝑔))
efmnd.4 𝐽 = (∏tβ€˜(𝐴 Γ— {𝒫 𝐴}))
Assertion
Ref Expression
efmnd (𝐴 ∈ 𝑉 β†’ 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
Distinct variable group:   𝑓,𝑔,𝐴
Allowed substitution hints:   𝐡(𝑓,𝑔)   + (𝑓,𝑔)   𝐺(𝑓,𝑔)   𝐽(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem efmnd
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efmnd.1 . 2 𝐺 = (EndoFMndβ€˜π΄)
2 elex 3461 . . 3 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ V)
3 ovexd 7388 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) ∈ V)
4 id 22 . . . . . . . 8 (𝑏 = (π‘Ž ↑m π‘Ž) β†’ 𝑏 = (π‘Ž ↑m π‘Ž))
5 id 22 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ π‘Ž = 𝐴)
65, 5oveq12d 7371 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) = (𝐴 ↑m 𝐴))
7 efmnd.2 . . . . . . . . 9 𝐡 = (𝐴 ↑m 𝐴)
86, 7eqtr4di 2794 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) = 𝐡)
94, 8sylan9eqr 2798 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ 𝑏 = 𝐡)
109opeq2d 4835 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐡⟩)
11 eqidd 2737 . . . . . . . . 9 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
129, 9, 11mpoeq123dv 7428 . . . . . . . 8 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑓 ∘ 𝑔)))
13 efmnd.3 . . . . . . . 8 + = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑓 ∘ 𝑔))
1412, 13eqtr4di 2794 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
1514opeq2d 4835 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+gβ€˜ndx), + ⟩)
16 simpl 483 . . . . . . . . . 10 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ π‘Ž = 𝐴)
17 pweq 4572 . . . . . . . . . . . 12 (π‘Ž = 𝐴 β†’ 𝒫 π‘Ž = 𝒫 𝐴)
1817sneqd 4596 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ {𝒫 π‘Ž} = {𝒫 𝐴})
1918adantr 481 . . . . . . . . . 10 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ {𝒫 π‘Ž} = {𝒫 𝐴})
2016, 19xpeq12d 5662 . . . . . . . . 9 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (π‘Ž Γ— {𝒫 π‘Ž}) = (𝐴 Γ— {𝒫 𝐴}))
2120fveq2d 6843 . . . . . . . 8 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž})) = (∏tβ€˜(𝐴 Γ— {𝒫 𝐴})))
22 efmnd.4 . . . . . . . 8 𝐽 = (∏tβ€˜(𝐴 Γ— {𝒫 𝐴}))
2321, 22eqtr4di 2794 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž})) = 𝐽)
2423opeq2d 4835 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩ = ⟨(TopSetβ€˜ndx), 𝐽⟩)
2510, 15, 24tpeq123d 4707 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
263, 25csbied 3891 . . . 4 (π‘Ž = 𝐴 β†’ ⦋(π‘Ž ↑m π‘Ž) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
27 df-efmnd 18671 . . . 4 EndoFMnd = (π‘Ž ∈ V ↦ ⦋(π‘Ž ↑m π‘Ž) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩})
28 tpex 7677 . . . 4 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩} ∈ V
2926, 27, 28fvmpt 6945 . . 3 (𝐴 ∈ V β†’ (EndoFMndβ€˜π΄) = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
302, 29syl 17 . 2 (𝐴 ∈ 𝑉 β†’ (EndoFMndβ€˜π΄) = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
311, 30eqtrid 2788 1 (𝐴 ∈ 𝑉 β†’ 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3443  β¦‹csb 3853  π’« cpw 4558  {csn 4584  {ctp 4588  βŸ¨cop 4590   Γ— cxp 5629   ∘ ccom 5635  β€˜cfv 6493  (class class class)co 7353   ∈ cmpo 7355   ↑m cmap 8761  ndxcnx 17057  Basecbs 17075  +gcplusg 17125  TopSetcts 17131  βˆtcpt 17312  EndoFMndcefmnd 18670
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-efmnd 18671
This theorem is referenced by:  efmndbas  18673  efmndtset  18681  efmndplusg  18682  symgvalstruct  19169  symgvalstructOLD  19170
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