Theorem termcfuncval 49190

Description: The value of a functor from a terminal category. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

Ref	Expression
diag1f1o.a	⊢ 𝐴 = (Base‘𝐶)
diag1f1o.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
termcfuncval.k	⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))
termcfuncval.b	⊢ 𝐵 = (Base‘𝐷)
termcfuncval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcfuncval.x	⊢ 𝑋 = ((1st ‘𝐾)‘𝑌)
termcfuncval.1	⊢ 1 = (Id‘𝐶)
termcfuncval.i	⊢ 𝐼 = (Id‘𝐷)

Assertion

Ref	Expression
termcfuncval	⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝐾 = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩))

Proof of Theorem termcfuncval

Step	Hyp	Ref	Expression
1		termcfuncval.x	. . 3 ⊢ 𝑋 = ((1st ‘𝐾)‘𝑌)
2		termcfuncval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
3		diag1f1o.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
4		termcfuncval.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))
5	4	func1st2nd 48927	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐶)(2nd ‘𝐾))
6	2, 3, 5	funcf1 17912	. . . 4 ⊢ (𝜑 → (1st ‘𝐾):𝐵⟶𝐴)
7		termcfuncval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	6, 7	ffvelcdmd 7104	. . 3 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) ∈ 𝐴)
9	1, 8	eqeltrid 2844	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
10		relfunc 17908	. . . 4 ⊢ Rel (𝐷 Func 𝐶)
11		1st2nd 8065	. . . 4 ⊢ ((Rel (𝐷 Func 𝐶) ∧ 𝐾 ∈ (𝐷 Func 𝐶)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
12	10, 4, 11	sylancr 587	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
13		diag1f1o.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ TermCat)
14	13, 2, 7	termcbas2 49153	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = {𝑌})
15	14	feq2d 6721	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐾):𝐵⟶𝐴 ↔ (1st ‘𝐾):{𝑌}⟶𝐴))
16	6, 15	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾):{𝑌}⟶𝐴)
17		fsn2g 7157	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → ((1st ‘𝐾):{𝑌}⟶𝐴 ↔ (((1st ‘𝐾)‘𝑌) ∈ 𝐴 ∧ (1st ‘𝐾) = {⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩})))
18	7, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → ((1st ‘𝐾):{𝑌}⟶𝐴 ↔ (((1st ‘𝐾)‘𝑌) ∈ 𝐴 ∧ (1st ‘𝐾) = {⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩})))
19	16, 18	mpbid 232	. . . . . 6 ⊢ (𝜑 → (((1st ‘𝐾)‘𝑌) ∈ 𝐴 ∧ (1st ‘𝐾) = {⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩}))
20	19	simprd 495	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾) = {⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩})
21	1	opeq2i 4876	. . . . . 6 ⊢ ⟨𝑌, 𝑋⟩ = ⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩
22	21	sneqi 4636	. . . . 5 ⊢ {⟨𝑌, 𝑋⟩} = {⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩}
23	20, 22	eqtr4di 2794	. . . 4 ⊢ (𝜑 → (1st ‘𝐾) = {⟨𝑌, 𝑋⟩})
24	2, 5	funcfn2 17915	. . . . . . 7 ⊢ (𝜑 → (2nd ‘𝐾) Fn (𝐵 × 𝐵))
25	14	sqxpeqd 5716	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐵) = ({𝑌} × {𝑌}))
26		xpsng 7158	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ({𝑌} × {𝑌}) = {⟨𝑌, 𝑌⟩})
27	7, 7, 26	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ({𝑌} × {𝑌}) = {⟨𝑌, 𝑌⟩})
28	25, 27	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = {⟨𝑌, 𝑌⟩})
29	28	fneq2d 6661	. . . . . . 7 ⊢ (𝜑 → ((2nd ‘𝐾) Fn (𝐵 × 𝐵) ↔ (2nd ‘𝐾) Fn {⟨𝑌, 𝑌⟩}))
30	24, 29	mpbid 232	. . . . . 6 ⊢ (𝜑 → (2nd ‘𝐾) Fn {⟨𝑌, 𝑌⟩})
31		opex 5468	. . . . . . 7 ⊢ ⟨𝑌, 𝑌⟩ ∈ V
32	31	fnsnb 7186	. . . . . 6 ⊢ ((2nd ‘𝐾) Fn {⟨𝑌, 𝑌⟩} ↔ (2nd ‘𝐾) = {⟨⟨𝑌, 𝑌⟩, ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)⟩})
33	30, 32	sylib 218	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐾) = {⟨⟨𝑌, 𝑌⟩, ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)⟩})
34		df-ov 7435	. . . . . . . 8 ⊢ (𝑌(2nd ‘𝐾)𝑌) = ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)
35		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
36		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	2, 35, 36, 5, 7, 7	funcf2 17914	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑌):(𝑌(Hom ‘𝐷)𝑌)⟶(((1st ‘𝐾)‘𝑌)(Hom ‘𝐶)((1st ‘𝐾)‘𝑌)))
38		termcfuncval.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (Id‘𝐷)
39	13, 2, 7, 7, 35, 38	termchom 49159	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌(Hom ‘𝐷)𝑌) = {(𝐼‘𝑌)})
40	39	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝐼‘𝑌)} = (𝑌(Hom ‘𝐷)𝑌))
41	1, 1	oveq12i 7444	. . . . . . . . . . . . . 14 ⊢ (𝑋(Hom ‘𝐶)𝑋) = (((1st ‘𝐾)‘𝑌)(Hom ‘𝐶)((1st ‘𝐾)‘𝑌))
42	41	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑋) = (((1st ‘𝐾)‘𝑌)(Hom ‘𝐶)((1st ‘𝐾)‘𝑌)))
43	40, 42	feq23d 6730	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑌):{(𝐼‘𝑌)}⟶(𝑋(Hom ‘𝐶)𝑋) ↔ (𝑌(2nd ‘𝐾)𝑌):(𝑌(Hom ‘𝐷)𝑌)⟶(((1st ‘𝐾)‘𝑌)(Hom ‘𝐶)((1st ‘𝐾)‘𝑌))))
44	37, 43	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑌):{(𝐼‘𝑌)}⟶(𝑋(Hom ‘𝐶)𝑋))
45		fvex 6918	. . . . . . . . . . . 12 ⊢ (𝐼‘𝑌) ∈ V
46	45	fsn2 7155	. . . . . . . . . . 11 ⊢ ((𝑌(2nd ‘𝐾)𝑌):{(𝐼‘𝑌)}⟶(𝑋(Hom ‘𝐶)𝑋) ↔ (((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌)) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ (𝑌(2nd ‘𝐾)𝑌) = {⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩}))
47	44, 46	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌)) ∈ (𝑋(Hom ‘𝐶)𝑋) ∧ (𝑌(2nd ‘𝐾)𝑌) = {⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩}))
48	47	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑌) = {⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩})
49		termcfuncval.1	. . . . . . . . . . . . 13 ⊢ 1 = (Id‘𝐶)
50	2, 38, 49, 5, 7	funcid 17916	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌)) = ( 1 ‘((1st ‘𝐾)‘𝑌)))
51	1	fveq2i 6908	. . . . . . . . . . . 12 ⊢ ( 1 ‘𝑋) = ( 1 ‘((1st ‘𝐾)‘𝑌))
52	50, 51	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌)) = ( 1 ‘𝑋))
53	52	opeq2d 4879	. . . . . . . . . 10 ⊢ (𝜑 → ⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩ = ⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩)
54	53	sneqd 4637	. . . . . . . . 9 ⊢ (𝜑 → {⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩} = {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩})
55	48, 54	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑌) = {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩})
56	34, 55	eqtr3id 2790	. . . . . . 7 ⊢ (𝜑 → ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩) = {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩})
57	56	opeq2d 4879	. . . . . 6 ⊢ (𝜑 → ⟨⟨𝑌, 𝑌⟩, ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)⟩ = ⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩)
58	57	sneqd 4637	. . . . 5 ⊢ (𝜑 → {⟨⟨𝑌, 𝑌⟩, ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)⟩} = {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩})
59	33, 58	eqtrd 2776	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩})
60	23, 59	opeq12d 4880	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩)
61	12, 60	eqtrd 2776	. 2 ⊢ (𝜑 → 𝐾 = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩)
62	9, 61	jca 511	1 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ 𝐾 = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {csn 4625  ⟨cop 4631   × cxp 5682  Rel wrel 5689   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  Basecbs 17248  Hom chom 17309  Idccid 17709   Func cfunc 17900  TermCatctermc 49144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869  df-ixp 8939  df-cat 17712  df-cid 17713  df-func 17904  df-thinc 49092  df-termc 49145

This theorem is referenced by:  diag1f1olem  49191