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Theorem efmnd 18026
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 3487 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovexd 7175 . . . . 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 7158 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎m 𝑎) = (𝐴m 𝐴))
7 efmnd.2 . . . . . . . . 9 𝐵 = (𝐴m 𝐴)
86, 7eqtr4di 2875 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎m 𝑎) = 𝐵)
94, 8sylan9eqr 2879 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑏 = 𝐵)
109opeq2d 4785 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 eqidd 2823 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑔) = (𝑓𝑔))
129, 9, 11mpoeq123dv 7213 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
13 efmnd.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
1412, 13eqtr4di 2875 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
1514opeq2d 4785 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16 simpl 486 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑎 = 𝐴)
17 pweq 4527 . . . . . . . . . . . 12 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
1817sneqd 4551 . . . . . . . . . . 11 (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
1918adantr 484 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
2016, 19xpeq12d 5563 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2120fveq2d 6656 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
22 efmnd.4 . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
2321, 22eqtr4di 2875 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
2423opeq2d 4785 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
2510, 15, 24tpeq123d 4658 . . . . 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 3891 . . . 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 18025 . . . 4 EndoFMnd = (𝑎 ∈ V ↦ (𝑎m 𝑎) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
28 tpex 7455 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
2926, 27, 28fvmpt 6750 . . 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, 30syl5eq 2869 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  csb 3855  𝒫 cpw 4511  {csn 4539  {ctp 4543  cop 4545   × cxp 5530  ccom 5536  cfv 6334  (class class class)co 7140  cmpo 7142  m cmap 8393  ndxcnx 16471  Basecbs 16474  +gcplusg 16556  TopSetcts 16562  tcpt 16703  EndoFMndcefmnd 18024
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-efmnd 18025
This theorem is referenced by:  efmndbas  18027  efmndtset  18035  efmndplusg  18036  symgvalstruct  18516
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