Theorem efmnd 18905

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 3509	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		ovexd 7483	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) ∈ V)
4		id 22	. . . . . . . 8 ⊢ (𝑏 = (𝑎 ↑m 𝑎) → 𝑏 = (𝑎 ↑m 𝑎))
5		id 22	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	5, 5	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) = (𝐴 ↑m 𝐴))
7		efmnd.2	. . . . . . . . 9 ⊢ 𝐵 = (𝐴 ↑m 𝐴)
8	6, 7	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) = 𝐵)
9	4, 8	sylan9eqr 2802	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → 𝑏 = 𝐵)
10	9	opeq2d 4904	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11		eqidd 2741	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
12	9, 9, 11	mpoeq123dv 7525	. . . . . . . 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 2798	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
15	14	opeq2d 4904	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16		simpl 482	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → 𝑎 = 𝐴)
17		pweq 4636	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
18	17	sneqd 4660	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
19	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
20	16, 19	xpeq12d 5731	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
21	20	fveq2d 6924	. . . . . . . 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 2798	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
24	23	opeq2d 4904	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
25	10, 15, 24	tpeq123d 4773	. . . . 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 3959	. . . 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 18904	. . . 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 7781	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
29	26, 27, 28	fvmpt 7029	. . 3 ⊢ (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
30	2, 29	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
31	1, 30	eqtrid 2792	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⦋csb 3921  𝒫 cpw 4622  {csn 4648  {ctp 4652  ⟨cop 4654   × cxp 5698   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ↑_m cmap 8884  ndxcnx 17240  Basecbs 17258  +_gcplusg 17311  TopSetcts 17317  ∏_tcpt 17498  EndoFMndcefmnd 18903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-efmnd 18904

This theorem is referenced by:  efmndbas  18906  efmndtset  18914  efmndplusg  18915  symgvalstruct  19438  symgvalstructOLD  19439