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Theorem efmnd 18680
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 3463 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovexd 7392 . . . . 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 7375 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎m 𝑎) = (𝐴m 𝐴))
7 efmnd.2 . . . . . . . . 9 𝐵 = (𝐴m 𝐴)
86, 7eqtr4di 2794 . . . . . . . 8 (𝑎 = 𝐴 → (𝑎m 𝑎) = 𝐵)
94, 8sylan9eqr 2798 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑏 = 𝐵)
109opeq2d 4837 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11 eqidd 2737 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑔) = (𝑓𝑔))
129, 9, 11mpoeq123dv 7432 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
13 efmnd.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
1412, 13eqtr4di 2794 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
1514opeq2d 4837 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16 simpl 483 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → 𝑎 = 𝐴)
17 pweq 4574 . . . . . . . . . . . 12 (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
1817sneqd 4598 . . . . . . . . . . 11 (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
1918adantr 481 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
2016, 19xpeq12d 5664 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2120fveq2d 6846 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
22 efmnd.4 . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
2321, 22eqtr4di 2794 . . . . . . 7 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
2423opeq2d 4837 . . . . . 6 ((𝑎 = 𝐴𝑏 = (𝑎m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
2510, 15, 24tpeq123d 4709 . . . . 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 3893 . . . 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 18679 . . . 4 EndoFMnd = (𝑎 ∈ V ↦ (𝑎m 𝑎) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
28 tpex 7681 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
2926, 27, 28fvmpt 6948 . . 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 2788 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855  𝒫 cpw 4560  {csn 4586  {ctp 4590  cop 4592   × cxp 5631  ccom 5637  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  ndxcnx 17065  Basecbs 17083  +gcplusg 17133  TopSetcts 17139  tcpt 17320  EndoFMndcefmnd 18678
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 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
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 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-efmnd 18679
This theorem is referenced by:  efmndbas  18681  efmndtset  18689  efmndplusg  18690  symgvalstruct  19178  symgvalstructOLD  19179
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