diag2f1o - Mathbox for Zhi Wang

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Theorem diag2f1o 50000

Description: If 𝐷 is terminal, the morphism part of a diagonal functor is bijective functions from hom-sets into sets of natural transformations. (Contributed by Zhi Wang, 21-Oct-2025.)

**Hypotheses**
Ref	Expression
diag2f1o.l	⊢ 𝐿 = (𝐶Δ_func𝐷)
diag2f1o.a	⊢ 𝐴 = (Base‘𝐶)
diag2f1o.h	⊢ 𝐻 = (Hom ‘𝐶)
diag2f1o.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
diag2f1o.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
diag2f1o.n	⊢ 𝑁 = (𝐷 Nat 𝐶)
diag2f1o.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
diag2f1o.c	⊢ (𝜑 → 𝐶 ∈ Cat)

**Assertion**
Ref	Expression
diag2f1o	⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1-onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))

Proof of Theorem diag2f1o Dummy variables 𝑓 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diag2f1o.l	. . 3 ⊢ 𝐿 = (𝐶Δ_func𝐷)
2		diag2f1o.a	. . 3 ⊢ 𝐴 = (Base‘𝐶)
3		eqid 2735	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		diag2f1o.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
5		diag2f1o.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		diag2f1o.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
7	6	termccd 49942	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		diag2f1o.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9		diag2f1o.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
10	3	istermc2 49938	. . . . . . 7 ⊢ (𝐷 ∈ TermCat ↔ (𝐷 ∈ ThinCat ∧ ∃!𝑧 𝑧 ∈ (Base‘𝐷)))
11	6, 10	sylib 218	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ ThinCat ∧ ∃!𝑧 𝑧 ∈ (Base‘𝐷)))
12	11	simprd 495	. . . . 5 ⊢ (𝜑 → ∃!𝑧 𝑧 ∈ (Base‘𝐷))
13		euex 2576	. . . . 5 ⊢ (∃!𝑧 𝑧 ∈ (Base‘𝐷) → ∃𝑧 𝑧 ∈ (Base‘𝐷))
14	12, 13	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ (Base‘𝐷))
15		n0 4283	. . . 4 ⊢ ((Base‘𝐷) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (Base‘𝐷))
16	14, 15	sylibr 234	. . 3 ⊢ (𝜑 → (Base‘𝐷) ≠ ∅)
17		diag2f1o.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐶)
18	1, 2, 3, 4, 5, 7, 8, 9, 16, 17	diag2f1 49772	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
19		f1f 6725	. . . 4 ⊢ ((𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)) → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)⟶(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)⟶(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
21	6, 3	termcbas 49943	. . . . . 6 ⊢ (𝜑 → ∃𝑧(Base‘𝐷) = {𝑧})
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) → ∃𝑧(Base‘𝐷) = {𝑧})
23		fveq2 6829	. . . . . . 7 ⊢ (𝑓 = (𝑚‘𝑧) → ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓) = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧)))
24	23	eqeq2d 2746	. . . . . 6 ⊢ (𝑓 = (𝑚‘𝑧) → (𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓) ↔ 𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧))))
25	8	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑋 ∈ 𝐴)
26	9	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑌 ∈ 𝐴)
27	6	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝐷 ∈ TermCat)
28		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
29		vsnid 4597	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
30		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → (Base‘𝐷) = {𝑧})
31	29, 30	eleqtrrid 2842	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑧 ∈ (Base‘𝐷))
32		eqid 2735	. . . . . . . 8 ⊢ (𝑚‘𝑧) = (𝑚‘𝑧)
33	1, 2, 4, 25, 26, 17, 27, 28, 3, 31, 32	diag2f1olem 49999	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → ((𝑚‘𝑧) ∈ (𝑋𝐻𝑌) ∧ 𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧))))
34	33	simpld 494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → (𝑚‘𝑧) ∈ (𝑋𝐻𝑌))
35	33	simprd 495	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧)))
36	24, 34, 35	rspcedvdw 3565	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
37	22, 36	exlimddv 1937	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
38	37	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
39		dffo3 7043	. . 3 ⊢ ((𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)) ↔ ((𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)⟶(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)) ∧ ∀𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓)))
40	20, 38, 39	sylanbrc 584	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
41		df-f1o 6494	. 2 ⊢ ((𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1-onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)) ↔ ((𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)) ∧ (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))))
42	18, 40, 41	sylanbrc 584	1 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1-onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∃!weu 2567 ≠ wne 2930 ∀wral 3049 ∃wrex 3059 ∅c0 4263 {csn 4557 ⟶wf 6483 –1-1→wf1 6484 –onto→wfo 6485 –1-1-onto→wf1o 6486 ‘cfv 6487 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17168 Hom chom 17220 Catccat 17619 Nat cnat 17900 Δ_funccdiag 18167 ThinCatcthinc 49880 TermCatctermc 49935

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8632 df-map 8764 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-struct 17106 df-slot 17141 df-ndx 17153 df-base 17169 df-hom 17233 df-cco 17234 df-cat 17623 df-cid 17624 df-func 17814 df-nat 17902 df-fuc 17903 df-xpc 18127 df-1stf 18128 df-curf 18169 df-diag 18171 df-thinc 49881 df-termc 49936

This theorem is referenced by: diagffth 50001

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