Theorem diag1 48977

Description: The constant functor of 𝑋. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
diag1.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
diag1.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diag1.d	⊢ (𝜑 → 𝐷 ∈ Cat)
diag1.a	⊢ 𝐴 = (Base‘𝐶)
diag1.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
diag1.k	⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
diag1.b	⊢ 𝐵 = (Base‘𝐷)
diag1.j	⊢ 𝐽 = (Hom ‘𝐷)
diag1.i	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
diag1	⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ 𝑋), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))⟩)

Distinct variable groups:   𝐵,𝑓,𝑦,𝑧   𝑓,𝐽   𝑓,𝐾,𝑦,𝑧   𝜑,𝑓,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑦,𝑧,𝑓)   𝐶(𝑦,𝑧,𝑓)   𝐷(𝑦,𝑧,𝑓)   1 (𝑦,𝑧,𝑓)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧,𝑓)   𝑋(𝑦,𝑧,𝑓)

Proof of Theorem diag1

Step	Hyp	Ref	Expression
1		relfunc 17903	. . 3 ⊢ Rel (𝐷 Func 𝐶)
2		diag1.l	. . . 4 ⊢ 𝐿 = (𝐶Δfunc𝐷)
3		diag1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		diag1.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		diag1.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
6		diag1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
7		diag1.k	. . . 4 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
8	2, 3, 4, 5, 6, 7	diag1cl 18283	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐶))
9		1st2nd 8060	. . 3 ⊢ ((Rel (𝐷 Func 𝐶) ∧ 𝐾 ∈ (𝐷 Func 𝐶)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
10	1, 8, 9	sylancr 587	. 2 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11		diag1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐷)
12		1st2ndbr 8063	. . . . . . 7 ⊢ ((Rel (𝐷 Func 𝐶) ∧ 𝐾 ∈ (𝐷 Func 𝐶)) → (1st ‘𝐾)(𝐷 Func 𝐶)(2nd ‘𝐾))
13	1, 8, 12	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐶)(2nd ‘𝐾))
14	11, 5, 13	funcf1 17907	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾):𝐵⟶𝐴)
15	14	feqmptd 6975	. . . 4 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ ((1st ‘𝐾)‘𝑦)))
16	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Cat)
17	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ Cat)
18	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑋 ∈ 𝐴)
19		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
20	2, 16, 17, 5, 18, 7, 11, 19	diag11 18284	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝐾)‘𝑦) = 𝑋)
21	20	mpteq2dva 5240	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ ((1st ‘𝐾)‘𝑦)) = (𝑦 ∈ 𝐵 ↦ 𝑋))
22	15, 21	eqtrd 2776	. . 3 ⊢ (𝜑 → (1st ‘𝐾) = (𝑦 ∈ 𝐵 ↦ 𝑋))
23	11, 13	funcfn2 17910	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐾) Fn (𝐵 × 𝐵))
24		fnov 7561	. . . . 5 ⊢ ((2nd ‘𝐾) Fn (𝐵 × 𝐵) ↔ (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑦(2nd ‘𝐾)𝑧)))
25	23, 24	sylib 218	. . . 4 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑦(2nd ‘𝐾)𝑧)))
26		diag1.j	. . . . . . . 8 ⊢ 𝐽 = (Hom ‘𝐷)
27		eqid 2736	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
28	13	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (1st ‘𝐾)(𝐷 Func 𝐶)(2nd ‘𝐾))
29		simp2 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
30		simp3 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
31	11, 26, 27, 28, 29, 30	funcf2 17909	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(2nd ‘𝐾)𝑧):(𝑦𝐽𝑧)⟶(((1st ‘𝐾)‘𝑦)(Hom ‘𝐶)((1st ‘𝐾)‘𝑧)))
32	31	feqmptd 6975	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(2nd ‘𝐾)𝑧) = (𝑓 ∈ (𝑦𝐽𝑧) ↦ ((𝑦(2nd ‘𝐾)𝑧)‘𝑓)))
33		simpl1 1192	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝜑)
34	33, 3	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝐶 ∈ Cat)
35	33, 4	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝐷 ∈ Cat)
36	33, 6	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝑋 ∈ 𝐴)
37	29	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝑦 ∈ 𝐵)
38		diag1.i	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
39	30	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝑧 ∈ 𝐵)
40		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → 𝑓 ∈ (𝑦𝐽𝑧))
41	2, 34, 35, 5, 36, 7, 11, 37, 26, 38, 39, 40	diag12 18285	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ 𝑓 ∈ (𝑦𝐽𝑧)) → ((𝑦(2nd ‘𝐾)𝑧)‘𝑓) = ( 1 ‘𝑋))
42	41	mpteq2dva 5240	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑓 ∈ (𝑦𝐽𝑧) ↦ ((𝑦(2nd ‘𝐾)𝑧)‘𝑓)) = (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))
43	32, 42	eqtrd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦(2nd ‘𝐾)𝑧) = (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))
44	43	mpoeq3dva 7508	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑦(2nd ‘𝐾)𝑧)) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋))))
45	25, 44	eqtrd 2776	. . 3 ⊢ (𝜑 → (2nd ‘𝐾) = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋))))
46	22, 45	opeq12d 4879	. 2 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ = ⟨(𝑦 ∈ 𝐵 ↦ 𝑋), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))⟩)
47	10, 46	eqtrd 2776	1 ⊢ (𝜑 → 𝐾 = ⟨(𝑦 ∈ 𝐵 ↦ 𝑋), (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐵 ↦ (𝑓 ∈ (𝑦𝐽𝑧) ↦ ( 1 ‘𝑋)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ⟨cop 4630   class class class wbr 5141   ↦ cmpt 5223   × cxp 5681  Rel wrel 5688   Fn wfn 6554  ‘cfv 6559  (class class class)co 7429   ∈ cmpo 7431  1^st c1st 8008  2^nd c2nd 8009  Basecbs 17243  Hom chom 17304  Catccat 17703  Idccid 17704   Func cfunc 17895  Δ_funccdiag 18253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751  ax-cnex 11207  ax-resscn 11208  ax-1cn 11209  ax-icn 11210  ax-addcl 11211  ax-addrcl 11212  ax-mulcl 11213  ax-mulrcl 11214  ax-mulcom 11215  ax-addass 11216  ax-mulass 11217  ax-distr 11218  ax-i2m1 11219  ax-1ne0 11220  ax-1rid 11221  ax-rnegex 11222  ax-rrecex 11223  ax-cnre 11224  ax-pre-lttri 11225  ax-pre-lttrn 11226  ax-pre-ltadd 11227  ax-pre-mulgt0 11228

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  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-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-riota 7386  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-1st 8010  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-1o 8502  df-er 8741  df-map 8864  df-ixp 8934  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-pnf 11293  df-mnf 11294  df-xr 11295  df-ltxr 11296  df-le 11297  df-sub 11490  df-neg 11491  df-nn 12263  df-2 12325  df-3 12326  df-4 12327  df-5 12328  df-6 12329  df-7 12330  df-8 12331  df-9 12332  df-n0 12523  df-z 12610  df-dec 12730  df-uz 12875  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17244  df-hom 17317  df-cco 17318  df-cat 17707  df-cid 17708  df-func 17899  df-nat 17987  df-fuc 17988  df-xpc 18213  df-1stf 18214  df-curf 18255  df-diag 18257

This theorem is referenced by: (None)