Theorem efmnd 18505

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 3449	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		ovexd 7304	. . . . 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 7287	. . . . . . . . 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 4817	. . . . . 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 7342	. . . . . . . 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 4817	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
16		simpl 483	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → 𝑎 = 𝐴)
17		pweq 4555	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → 𝒫 𝑎 = 𝒫 𝐴)
18	17	sneqd 4579	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → {𝒫 𝑎} = {𝒫 𝐴})
19	18	adantr 481	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → {𝒫 𝑎} = {𝒫 𝐴})
20	16, 19	xpeq12d 5620	. . . . . . . . 9 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
21	20	fveq2d 6773	. . . . . . . 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 4817	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = (𝑎 ↑m 𝑎)) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
25	10, 15, 24	tpeq123d 4690	. . . . 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 3875	. . . 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 18504	. . . 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 7589	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
29	26, 27, 28	fvmpt 6870	. . 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 396   = wceq 1542   ∈ wcel 2110  Vcvv 3431  ⦋csb 3837  𝒫 cpw 4539  {csn 4567  {ctp 4571  ⟨cop 4573   × cxp 5587   ∘ ccom 5593  ‘cfv 6431  (class class class)co 7269   ∈ cmpo 7271   ↑_m cmap 8596  ndxcnx 16890  Basecbs 16908  +_gcplusg 16958  TopSetcts 16964  ∏_tcpt 17145  EndoFMndcefmnd 18503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6389  df-fun 6433  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-efmnd 18504

This theorem is referenced by:  efmndbas  18506  efmndtset  18514  efmndplusg  18515  symgvalstruct  19000  symgvalstructOLD  19001