diag2f1o - Mathbox for Zhi Wang

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Theorem diag2f1o 49389

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 49332	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		diag2f1o.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9		diag2f1o.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
10	3	istermc2 49328	. . . . . . 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 2577	. . . . 5 ⊢ (∃!𝑧 𝑧 ∈ (Base‘𝐷) → ∃𝑧 𝑧 ∈ (Base‘𝐷))
14	12, 13	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ (Base‘𝐷))
15		n0 4333	. . . 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 49187	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
19		f1f 6779	. . . 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 49333	. . . . . 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 6881	. . . . . . 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 4644	. . . . . . . . 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 2736	. . . . . . . 8 ⊢ (𝑚‘𝑧) = (𝑚‘𝑧)
33	1, 2, 4, 25, 26, 17, 27, 28, 3, 31, 32	diag2f1olem 49388	. . . . . . 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 3609	. . . . 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 1935	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) → ∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
38	37	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
39		dffo3 7097	. . 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 6543	. 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 1540 ∃wex 1779 ∈ wcel 2109 ∃!weu 2568 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ∅c0 4313 {csn 4606 ⟶wf 6532 –1-1→wf1 6533 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 Basecbs 17233 Hom chom 17287 Catccat 17681 Nat cnat 17962 Δ_funccdiag 18229 ThinCatcthinc 49270 TermCatctermc 49325

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-hom 17300 df-cco 17301 df-cat 17685 df-cid 17686 df-func 17876 df-nat 17964 df-fuc 17965 df-xpc 18189 df-1stf 18190 df-curf 18231 df-diag 18233 df-thinc 49271 df-termc 49326

This theorem is referenced by: diagffth 49390

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