Theorem diag1f1o 49386

Description: The object part of the diagonal functor is a bijection if 𝐷 is terminal. So any functor from a terminal category is one-to-one correspondent to an object of the target base. (Contributed by Zhi Wang, 21-Oct-2025.)

Hypotheses

Ref	Expression
diag1f1o.a	⊢ 𝐴 = (Base‘𝐶)
diag1f1o.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
diag1f1o.c	⊢ (𝜑 → 𝐶 ∈ Cat)
diag1f1o.l	⊢ 𝐿 = (𝐶Δfunc𝐷)

Assertion

Ref	Expression
diag1f1o	⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))

Proof of Theorem diag1f1o

Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diag1f1o.l	. . 3 ⊢ 𝐿 = (𝐶Δfunc𝐷)
2		diag1f1o.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		diag1f1o.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
4	3	termccd 49332	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		diag1f1o.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
6		eqid 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
7	6	istermc2 49328	. . . . . . 7 ⊢ (𝐷 ∈ TermCat ↔ (𝐷 ∈ ThinCat ∧ ∃!𝑦 𝑦 ∈ (Base‘𝐷)))
8	3, 7	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ ThinCat ∧ ∃!𝑦 𝑦 ∈ (Base‘𝐷)))
9	8	simprd 495	. . . . 5 ⊢ (𝜑 → ∃!𝑦 𝑦 ∈ (Base‘𝐷))
10		euex 2577	. . . . 5 ⊢ (∃!𝑦 𝑦 ∈ (Base‘𝐷) → ∃𝑦 𝑦 ∈ (Base‘𝐷))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (Base‘𝐷))
12		n0 4333	. . . 4 ⊢ ((Base‘𝐷) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (Base‘𝐷))
13	11, 12	sylibr 234	. . 3 ⊢ (𝜑 → (Base‘𝐷) ≠ ∅)
14	1, 2, 4, 5, 6, 13	diag1f1 49185	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
15		f1f 6779	. . . 4 ⊢ ((1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶) → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
17	3, 6	termcbas 49333	. . . . . 6 ⊢ (𝜑 → ∃𝑦(Base‘𝐷) = {𝑦})
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑦(Base‘𝐷) = {𝑦})
19		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦)))
20	19	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → (𝑘 = ((1st ‘𝐿)‘𝑥) ↔ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
21	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝐷 ∈ TermCat)
22		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 ∈ (𝐷 Func 𝐶))
23		vsnid 4644	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
24		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → (Base‘𝐷) = {𝑦})
25	23, 24	eleqtrrid 2842	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑦 ∈ (Base‘𝐷))
26		eqid 2736	. . . . . . . 8 ⊢ ((1st ‘𝑘)‘𝑦) = ((1st ‘𝑘)‘𝑦)
27	5, 21, 22, 6, 25, 26, 1	diag1f1olem 49385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → (((1st ‘𝑘)‘𝑦) ∈ 𝐴 ∧ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
28	27	simpld 494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ((1st ‘𝑘)‘𝑦) ∈ 𝐴)
29	27	simprd 495	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦)))
30	20, 28, 29	rspcedvdw 3609	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
31	18, 30	exlimddv 1935	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
32	31	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (𝐷 Func 𝐶)∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
33		dffo3 7097	. . 3 ⊢ ((1st ‘𝐿):𝐴–onto→(𝐷 Func 𝐶) ↔ ((1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶) ∧ ∀𝑘 ∈ (𝐷 Func 𝐶)∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥)))
34	16, 32, 33	sylanbrc 583	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–onto→(𝐷 Func 𝐶))
35		df-f1o 6543	. 2 ⊢ ((1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶) ↔ ((1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶) ∧ (1st ‘𝐿):𝐴–onto→(𝐷 Func 𝐶)))
36	14, 34, 35	sylanbrc 583	1 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2568   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  ∅c0 4313  {csn 4606  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  Basecbs 17233  Catccat 17681   Func cfunc 17872  Δ_funccdiag 18229  ThinCatcthinc 49270  TermCatctermc 49325

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-cat 17685  df-cid 17686  df-func 17876  df-nat 17964  df-fuc 17965  df-xpc 18189  df-1stf 18190  df-curf 18231  df-diag 18233  df-thinc 49271  df-termc 49326

This theorem is referenced by:  diagciso  49391