Theorem diag1f1o 50108

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 50053	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
5		diag1f1o.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
6		eqid 2761	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
7	6	istermc2 50049	. . . . . . 7 ⊢ (𝐷 ∈ TermCat ↔ (𝐷 ∈ ThinCat ∧ ∃!𝑦 𝑦 ∈ (Base‘𝐷)))
8	3, 7	sylib 220	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ ThinCat ∧ ∃!𝑦 𝑦 ∈ (Base‘𝐷)))
9	8	simprd 499	. . . . 5 ⊢ (𝜑 → ∃!𝑦 𝑦 ∈ (Base‘𝐷))
10		euex 2603	. . . . 5 ⊢ (∃!𝑦 𝑦 ∈ (Base‘𝐷) → ∃𝑦 𝑦 ∈ (Base‘𝐷))
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑦 𝑦 ∈ (Base‘𝐷))
12		n0 4305	. . . 4 ⊢ ((Base‘𝐷) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (Base‘𝐷))
13	11, 12	sylibr 236	. . 3 ⊢ (𝜑 → (Base‘𝐷) ≠ ∅)
14	1, 2, 4, 5, 6, 13	diag1f1 49881	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1→(𝐷 Func 𝐶))
15		f1f 6754	. . . 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 50054	. . . . . 6 ⊢ (𝜑 → ∃𝑦(Base‘𝐷) = {𝑦})
18	17	adantr 484	. . . . 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 2772	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑘)‘𝑦) → (𝑘 = ((1st ‘𝐿)‘𝑥) ↔ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
21	3	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝐷 ∈ TermCat)
22		simplr 778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 ∈ (𝐷 Func 𝐶))
23		vsnid 4621	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
24		simpr 488	. . . . . . . . 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 50107	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → (((1st ‘𝑘)‘𝑦) ∈ 𝐴 ∧ 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦))))
28	27	simpld 498	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ((1st ‘𝑘)‘𝑦) ∈ 𝐴)
29	27	simprd 499	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → 𝑘 = ((1st ‘𝐿)‘((1st ‘𝑘)‘𝑦)))
30	20, 28, 29	rspcedvdw 3584	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) ∧ (Base‘𝐷) = {𝑦}) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
31	18, 30	exlimddv 1954	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐷 Func 𝐶)) → ∃𝑥 ∈ 𝐴 𝑘 = ((1st ‘𝐿)‘𝑥))
32	31	ralrimiva 3153	. . 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 592	. 2 ⊢ (𝜑 → (1st ‘𝐿):𝐴–onto→(𝐷 Func 𝐶))
35		df-f1o 6522	. 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 592	1 ⊢ (𝜑 → (1st ‘𝐿):𝐴–1-1-onto→(𝐷 Func 𝐶))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃!weu 2594   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  ∅c0 4285  {csn 4581  ⟶wf 6511  –1-1→wf1 6512  –onto→wfo 6513  –1-1-onto→wf1o 6514  ‘cfv 6515  (class class class)co 7390  1^st c1st 7962  Basecbs 17226  Catccat 17677   Func cfunc 17868  Δ_funccdiag 18225  ThinCatcthinc 49991  TermCatctermc 50046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11213  df-mnf 11214  df-xr 11215  df-ltxr 11216  df-le 11217  df-sub 11411  df-neg 11412  df-nn 12206  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12477  df-z 12564  df-dec 12684  df-uz 12835  df-fz 13508  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17291  df-cco 17292  df-cat 17681  df-cid 17682  df-func 17872  df-nat 17960  df-fuc 17961  df-xpc 18185  df-1stf 18186  df-curf 18227  df-diag 18229  df-thinc 49992  df-termc 50047

This theorem is referenced by:  diagciso  50113  lmdran  50245  cmdlan  50246