Theorem diag1f1o 49527

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 49472	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		diag1f1o.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
6		eqid 2730	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
7	6	istermc2 49468	. . . . . . 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 2571	. . . . 5 ⊢ (∃!𝑦 𝑦 ∈ (Base‘𝐷) → ∃𝑦 𝑦 ∈ (Base‘𝐷))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (Base‘𝐷))
12		n0 4319	. . . 4 ⊢ ((Base‘𝐷) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (Base‘𝐷))
13	11, 12	sylibr 234	. . 3 ⊢ (𝜑 → (Base‘𝐷) ≠ ∅)
14	1, 2, 4, 5, 6, 13	diag1f1 49300	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
15		f1f 6759	. . . 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 49473	. . . . . 6 ⊢ (𝜑 → ∃𝑦(Base‘𝐷) = {𝑦})
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑦(Base‘𝐷) = {𝑦})
19		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦)))
20	19	eqeq2d 2741	. . . . . 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 4630	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
24		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → (Base‘𝐷) = {𝑦})
25	23, 24	eleqtrrid 2836	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑦 ∈ (Base‘𝐷))
26		eqid 2730	. . . . . . . 8 ⊢ ((1st ‘𝑘)‘𝑦) = ((1st ‘𝑘)‘𝑦)
27	5, 21, 22, 6, 25, 26, 1	diag1f1olem 49526	. . . . . . 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 3594	. . . . 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 3126	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (𝐷 Func 𝐶)∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
33		dffo3 7077	. . 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 6521	. 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 2562   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  ∅c0 4299  {csn 4592  ⟶wf 6510  –1-1→wf1 6511  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  Basecbs 17186  Catccat 17632   Func cfunc 17823  Δ_funccdiag 18180  ThinCatcthinc 49410  TermCatctermc 49465

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-slot 17159  df-ndx 17171  df-base 17187  df-hom 17251  df-cco 17252  df-cat 17636  df-cid 17637  df-func 17827  df-nat 17915  df-fuc 17916  df-xpc 18140  df-1stf 18141  df-curf 18182  df-diag 18184  df-thinc 49411  df-termc 49466

This theorem is referenced by:  diagciso  49532  lmdran  49664  cmdlan  49665