Theorem efmnd 18953

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 = (𝐴 ↑m 𝐴)
efmnd.3	⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
efmnd.4	⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))

Assertion

Ref	Expression
efmnd	⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(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.

Step	Hyp	Ref	Expression
1		efmnd.1	. 2 ⊢ 𝐺 = (EndoFMnd‘𝐴)
2		elex 3478	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		ovexd 7451	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) ∈ V)
4		id 23	. . . . . . . 8 ⊢ (𝑏 = (𝑎 ↑m 𝑎) → 𝑏 = (𝑎 ↑m 𝑎))
5		id 23	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	5, 5	oveq12d 7434	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) = (𝐴 ↑m 𝐴))
7		efmnd.2	. . . . . . . . 9 ⊢ 𝐵 = (𝐴 ↑m 𝐴)
8	6, 7	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) = 𝐵)
9	4, 8	sylan9eqr 2822	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → 𝑏 = 𝐵)
10	9	opeq2d 4847	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11		eqidd 2766	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
12	9, 9, 11	mpoeq123dv 7491	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)))
13		efmnd.3	. . . . . . . 8 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
14	12, 13	eqtr4di 2818	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
15	14	opeq2d 4847	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16		simpl 488	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → 𝑎 = 𝐴)
17		pweq 4578	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
18	17	sneqd 4603	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
19	18	adantr 486	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
20	16, 19	xpeq12d 5694	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
21	20	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
22		efmnd.4	. . . . . . . 8 ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
23	21, 22	eqtr4di 2818	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
24	23	opeq2d 4847	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
25	10, 15, 24	tpeq123d 4716	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
26	3, 25	csbied 3890	. . . 4 ⊢ (𝑎 = 𝐴 → ⦋(𝑎 ↑m 𝑎) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
27		df-efmnd 18952	. . . 4 ⊢ EndoFMnd = (𝑎 ∈ V ↦ ⦋(𝑎 ↑m 𝑎) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
28		tpex 7749	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
29	26, 27, 28	fvmpt 6993	. . 3 ⊢ (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
30	2, 29	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
31	1, 30	eqtrid 2812	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⦋csb 3854  𝒫 cpw 4564  {csn 4591  {ctp 4595  ⟨cop 4597   × cxp 5661   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418   ↑_m cmap 8826  ndxcnx 17271  Basecbs 17287  +_gcplusg 17328  TopSetcts 17334  ∏_tcpt 17509  EndoFMndcefmnd 18951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-efmnd 18952

This theorem is used by:  efmndbas  18954  efmndtset  18962  efmndplusg  18963  symgvalstruct  19491