Theorem diag1f1o 50257

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 50202	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		diag1f1o.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
6		eqid 2761	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
7	6	istermc2 50198	. . . . . . 7 ⊢ (𝐷 ∈ TermCat ↔ (𝐷 ∈ ThinCat ∧ ∃!𝑦 𝑦 ∈ (Base‘𝐷)))
8	3, 7	sylib 221	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ ThinCat ∧ ∃!𝑦 𝑦 ∈ (Base‘𝐷)))
9	8	simprd 500	. . . . 5 ⊢ (𝜑 → ∃!𝑦 𝑦 ∈ (Base‘𝐷))
10		euex 2603	. . . . 5 ⊢ (∃!𝑦 𝑦 ∈ (Base‘𝐷) → ∃𝑦 𝑦 ∈ (Base‘𝐷))
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (Base‘𝐷))
12		n0 4306	. . . 4 ⊢ ((Base‘𝐷) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (Base‘𝐷))
13	11, 12	sylibr 237	. . 3 ⊢ (𝜑 → (Base‘𝐷) ≠ ∅)
14	1, 2, 4, 5, 6, 13	diag1f1 50030	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
15		f1f 6774	. . . 4 ⊢ ((1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶) → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → (1st ‘𝐿):𝐴⟶(𝐷 Func 𝐶))
17	3, 6	termcbas 50203	. . . . . 6 ⊢ (𝜑 → ∃𝑦(Base‘𝐷) = {𝑦})
18	17	adantr 485	. . . . 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 2772	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → (𝑘 = ((1st ‘𝐿)‘𝑥) ↔ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
21	3	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝐷 ∈ TermCat)
22		simplr 780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 ∈ (𝐷 Func 𝐶))
23		vsnid 4628	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
24		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → (Base‘𝐷) = {𝑦})
25	23, 24	eleqtrrid 2868	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑦 ∈ (Base‘𝐷))
26		eqid 2761	. . . . . . . 8 ⊢ ((1st ‘𝑘)‘𝑦) = ((1st ‘𝑘)‘𝑦)
27	5, 21, 22, 6, 25, 26, 1	diag1f1olem 50256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → (((1st ‘𝑘)‘𝑦) ∈ 𝐴 ∧ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
28	27	simpld 499	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ((1st ‘𝑘)‘𝑦) ∈ 𝐴)
29	27	simprd 500	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦)))
30	20, 28, 29	rspcedvdw 3583	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
31	18, 30	exlimddv 1963	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
32	31	ralrimiva 3155	. . 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 594	. 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 594	1 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141  ∃!weu 2594   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  ∅c0 4285  {csn 4588  ⟶wf 6532  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  Basecbs 17268  Catccat 17719   Func cfunc 17910  Δ_funccdiag 18267  ThinCatcthinc 50140  TermCatctermc 50195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-er 8693  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-struct 17206  df-slot 17241  df-ndx 17253  df-base 17269  df-hom 17333  df-cco 17334  df-cat 17723  df-cid 17724  df-func 17914  df-nat 18002  df-fuc 18003  df-xpc 18227  df-1stf 18228  df-curf 18269  df-diag 18271  df-thinc 50141  df-termc 50196

This theorem is referenced by:  diagciso  50262  lmdran  50394  cmdlan  50395