diag2f1o - Mathbox for Zhi Wang

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Theorem diag2f1o 49195

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 2736	. . 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 49151	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		diag2f1o.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9		diag2f1o.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
10	3	istermc2 49147	. . . . . . 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 4352	. . . 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 49027	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
19		f1f 6803	. . . 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 49152	. . . . . 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 6905	. . . . . . 7 ⊢ (𝑓 = (𝑚‘𝑧) → ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓) = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧)))
24	23	eqeq2d 2747	. . . . . 6 ⊢ (𝑓 = (𝑚‘𝑧) → (𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓) ↔ 𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧))))
25	8	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑋 ∈ 𝐴)
26	9	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑌 ∈ 𝐴)
27	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝐷 ∈ TermCat)
28		simplr 768	. . . . . . . 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 4662	. . . . . . . . 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 2847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑧 ∈ (Base‘𝐷))
32		eqid 2736	. . . . . . . 8 ⊢ (𝑚‘𝑧) = (𝑚‘𝑧)
33	1, 2, 4, 25, 26, 17, 27, 28, 3, 31, 32	diag2f1olem 49194	. . . . . . 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 3624	. . . . 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 1934	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
38	37	ralrimiva 3145	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
39		dffo3 7121	. . 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 583	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–onto→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
41		df-f1o 6567	. 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 583	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 1539 ∃wex 1778 ∈ wcel 2107 ∃!weu 2567 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∅c0 4332 {csn 4625 ⟶wf 6556 –1-1→wf1 6557 –onto→wfo 6558 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 1^st c1st 8013 2^nd c2nd 8014 Basecbs 17248 Hom chom 17309 Catccat 17708 Nat cnat 17990 Δ_funccdiag 18258 ThinCatcthinc 49091 TermCatctermc 49144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-hom 17322 df-cco 17323 df-cat 17712 df-cid 17713 df-func 17904 df-nat 17992 df-fuc 17993 df-xpc 18218 df-1stf 18219 df-curf 18260 df-diag 18262 df-thinc 49092 df-termc 49145

This theorem is referenced by: diagffth 49196

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