Theorem termcfuncval 50004

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 49548	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐶)(2nd ‘𝐾))
6	2, 3, 5	funcf1 17822	. . . 4 ⊢ (𝜑 → (1st ‘𝐾):𝐵⟶𝐴)
7		termcfuncval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	6, 7	ffvelcdmd 7029	. . 3 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) ∈ 𝐴)
9	1, 8	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
10		relfunc 17818	. . . 4 ⊢ Rel (𝐷 Func 𝐶)
11		1st2nd 7983	. . . 4 ⊢ ((Rel (𝐷 Func 𝐶) ∧ 𝐾 ∈ (𝐷 Func 𝐶)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
12	10, 4, 11	sylancr 588	. . 3 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
13		diag1f1o.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ TermCat)
14	13, 2, 7	termcbas2 49954	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = {𝑌})
15	14	feq2d 6644	. . . . . . . 8 ⊢ (𝜑 → ((1st ‘𝐾):𝐵⟶𝐴 ↔ (1st ‘𝐾):{𝑌}⟶𝐴))
16	6, 15	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (1st ‘𝐾):{𝑌}⟶𝐴)
17		fsn2g 7083	. . . . . . . 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 4821	. . . . . 6 ⊢ ⟨𝑌, 𝑋⟩ = ⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩
22	21	sneqi 4579	. . . . 5 ⊢ {⟨𝑌, 𝑋⟩} = {⟨𝑌, ((1st ‘𝐾)‘𝑌)⟩}
23	20, 22	eqtr4di 2790	. . . 4 ⊢ (𝜑 → (1st ‘𝐾) = {⟨𝑌, 𝑋⟩})
24	2, 5	funcfn2 17825	. . . . . . 7 ⊢ (𝜑 → (2nd ‘𝐾) Fn (𝐵 × 𝐵))
25	14	sqxpeqd 5654	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 × 𝐵) = ({𝑌} × {𝑌}))
26		xpsng 7084	. . . . . . . . . 10 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ({𝑌} × {𝑌}) = {⟨𝑌, 𝑌⟩})
27	7, 7, 26	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ({𝑌} × {𝑌}) = {⟨𝑌, 𝑌⟩})
28	25, 27	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = {⟨𝑌, 𝑌⟩})
29	28	fneq2d 6584	. . . . . . 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 5409	. . . . . . 7 ⊢ ⟨𝑌, 𝑌⟩ ∈ V
32	31	fnsnb 7111	. . . . . 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 7361	. . . . . . . 8 ⊢ (𝑌(2nd ‘𝐾)𝑌) = ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)
35		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
36		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	2, 35, 36, 5, 7, 7	funcf2 17824	. . . . . . . . . . . 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 49960	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌(Hom ‘𝐷)𝑌) = {(𝐼‘𝑌)})
40	39	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(𝐼‘𝑌)} = (𝑌(Hom ‘𝐷)𝑌))
41	1, 1	oveq12i 7370	. . . . . . . . . . . . . 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 6655	. . . . . . . . . . . 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 6845	. . . . . . . . . . . 12 ⊢ (𝐼‘𝑌) ∈ V
46	45	fsn2 7081	. . . . . . . . . . 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 17826	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌)) = ( 1 ‘((1st ‘𝐾)‘𝑌)))
51	1	fveq2i 6835	. . . . . . . . . . . 12 ⊢ ( 1 ‘𝑋) = ( 1 ‘((1st ‘𝐾)‘𝑌))
52	50, 51	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌)) = ( 1 ‘𝑋))
53	52	opeq2d 4824	. . . . . . . . . 10 ⊢ (𝜑 → ⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩ = ⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩)
54	53	sneqd 4580	. . . . . . . . 9 ⊢ (𝜑 → {⟨(𝐼‘𝑌), ((𝑌(2nd ‘𝐾)𝑌)‘(𝐼‘𝑌))⟩} = {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩})
55	48, 54	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → (𝑌(2nd ‘𝐾)𝑌) = {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩})
56	34, 55	eqtr3id 2786	. . . . . . 7 ⊢ (𝜑 → ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩) = {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩})
57	56	opeq2d 4824	. . . . . 6 ⊢ (𝜑 → ⟨⟨𝑌, 𝑌⟩, ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)⟩ = ⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩)
58	57	sneqd 4580	. . . . 5 ⊢ (𝜑 → {⟨⟨𝑌, 𝑌⟩, ((2nd ‘𝐾)‘⟨𝑌, 𝑌⟩)⟩} = {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩})
59	33, 58	eqtrd 2772	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩})
60	23, 59	opeq12d 4825	. . 3 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ = ⟨{⟨𝑌, 𝑋⟩}, {⟨⟨𝑌, 𝑌⟩, {⟨(𝐼‘𝑌), ( 1 ‘𝑋)⟩}⟩}⟩)
61	12, 60	eqtrd 2772	. 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 1542   ∈ wcel 2114  {csn 4568  ⟨cop 4574   × cxp 5620  Rel wrel 5627   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Basecbs 17168  Hom chom 17220  Idccid 17620   Func cfunc 17810  TermCatctermc 49944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8766  df-ixp 8837  df-cat 17623  df-cid 17624  df-func 17814  df-thinc 49890  df-termc 49945

This theorem is referenced by:  diag1f1olem  50005