Theorem diag1f1o 49816

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 49761	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		diag1f1o.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
6		eqid 2735	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
7	6	istermc2 49757	. . . . . . 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 2576	. . . . 5 ⊢ (∃!𝑦 𝑦 ∈ (Base‘𝐷) → ∃𝑦 𝑦 ∈ (Base‘𝐷))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (Base‘𝐷))
12		n0 4304	. . . 4 ⊢ ((Base‘𝐷) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (Base‘𝐷))
13	11, 12	sylibr 234	. . 3 ⊢ (𝜑 → (Base‘𝐷) ≠ ∅)
14	1, 2, 4, 5, 6, 13	diag1f1 49589	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
15		f1f 6729	. . . 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 49762	. . . . . 6 ⊢ (𝜑 → ∃𝑦(Base‘𝐷) = {𝑦})
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑦(Base‘𝐷) = {𝑦})
19		fveq2 6833	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → ((1st ‘𝐿)‘𝑥) = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦)))
20	19	eqeq2d 2746	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → (𝑘 = ((1st ‘𝐿)‘𝑥) ↔ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
21	3	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝐷 ∈ TermCat)
22		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 ∈ (𝐷 Func 𝐶))
23		vsnid 4619	. . . . . . . . 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 2735	. . . . . . . 8 ⊢ ((1st ‘𝑘)‘𝑦) = ((1st ‘𝑘)‘𝑦)
27	5, 21, 22, 6, 25, 26, 1	diag1f1olem 49815	. . . . . . 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 3578	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
31	18, 30	exlimddv 1937	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
32	31	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (𝐷 Func 𝐶)∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
33		dffo3 7047	. . 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 584	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–onto→(𝐷 Func 𝐶))
35		df-f1o 6498	. 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 584	1 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2567   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  ∅c0 4284  {csn 4579  ⟶wf 6487  –1-1→wf1 6488  –onto→wfo 6489  –1-1-onto→wf1o 6490  ‘cfv 6491  (class class class)co 7358  1^st c1st 7931  Basecbs 17138  Catccat 17589   Func cfunc 17780  Δ_funccdiag 18137  ThinCatcthinc 49699  TermCatctermc 49754

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8767  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-fz 13426  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-hom 17203  df-cco 17204  df-cat 17593  df-cid 17594  df-func 17784  df-nat 17872  df-fuc 17873  df-xpc 18097  df-1stf 18098  df-curf 18139  df-diag 18141  df-thinc 49700  df-termc 49755

This theorem is referenced by:  diagciso  49821  lmdran  49953  cmdlan  49954