Theorem efmnd 18424

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 3440	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		ovexd 7290	. . . . 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 7273	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) = (𝐴 ↑m 𝐴))
7		efmnd.2	. . . . . . . . 9 ⊢ 𝐵 = (𝐴 ↑m 𝐴)
8	6, 7	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ↑m 𝑎) = 𝐵)
9	4, 8	sylan9eqr 2801	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → 𝑏 = 𝐵)
10	9	opeq2d 4808	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
11		eqidd 2739	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
12	9, 9, 11	mpoeq123dv 7328	. . . . . . . 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 2797	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
15	14	opeq2d 4808	. . . . . 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 4546	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
18	17	sneqd 4570	. . . . . . . . . . 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 5611	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
21	20	fveq2d 6760	. . . . . . . 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 2797	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
24	23	opeq2d 4808	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
25	10, 15, 24	tpeq123d 4681	. . . . 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 3866	. . . 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 18423	. . . 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 7575	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
29	26, 27, 28	fvmpt 6857	. . 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 2790	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⦋csb 3828  𝒫 cpw 4530  {csn 4558  {ctp 4562  ⟨cop 4564   × cxp 5578   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ↑_m cmap 8573  ndxcnx 16822  Basecbs 16840  +_gcplusg 16888  TopSetcts 16894  ∏_tcpt 17066  EndoFMndcefmnd 18422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-efmnd 18423

This theorem is referenced by:  efmndbas  18425  efmndtset  18433  efmndplusg  18434  symgvalstruct  18919  symgvalstructOLD  18920