diag2f1o - Mathbox for Zhi Wang

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Theorem diag2f1o 49283

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 2734	. . 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 49226	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		diag2f1o.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9		diag2f1o.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
10	3	istermc2 49222	. . . . . . 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 2575	. . . . 5 ⊢ (∃!𝑧 𝑧 ∈ (Base‘𝐷) → ∃𝑧 𝑧 ∈ (Base‘𝐷))
14	12, 13	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ (Base‘𝐷))
15		n0 4326	. . . 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 49083	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
19		f1f 6771	. . . 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 49227	. . . . . 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 6873	. . . . . . 7 ⊢ (𝑓 = (𝑚‘𝑧) → ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓) = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧)))
24	23	eqeq2d 2745	. . . . . 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 4637	. . . . . . . . 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 2840	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑧 ∈ (Base‘𝐷))
32		eqid 2734	. . . . . . . 8 ⊢ (𝑚‘𝑧) = (𝑚‘𝑧)
33	1, 2, 4, 25, 26, 17, 27, 28, 3, 31, 32	diag2f1olem 49282	. . . . . . 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 3602	. . . . 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 3130	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
39		dffo3 7089	. . 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 6535	. 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 2566 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 ∅c0 4306 {csn 4599 ⟶wf 6524 –1-1→wf1 6525 –onto→wfo 6526 –1-1-onto→wf1o 6527 ‘cfv 6528 (class class class)co 7400 1^st c1st 7981 2^nd c2nd 7982 Basecbs 17215 Hom chom 17269 Catccat 17663 Nat cnat 17944 Δ_funccdiag 18211 ThinCatcthinc 49166 TermCatctermc 49219

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 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199

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 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-tp 4604 df-op 4606 df-uni 4882 df-iun 4967 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7857 df-1st 7983 df-2nd 7984 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-1o 8475 df-er 8714 df-map 8837 df-ixp 8907 df-en 8955 df-dom 8956 df-sdom 8957 df-fin 8958 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-nn 12234 df-2 12296 df-3 12297 df-4 12298 df-5 12299 df-6 12300 df-7 12301 df-8 12302 df-9 12303 df-n0 12495 df-z 12582 df-dec 12702 df-uz 12846 df-fz 13515 df-struct 17153 df-slot 17188 df-ndx 17200 df-base 17216 df-hom 17282 df-cco 17283 df-cat 17667 df-cid 17668 df-func 17858 df-nat 17946 df-fuc 17947 df-xpc 18171 df-1stf 18172 df-curf 18213 df-diag 18215 df-thinc 49167 df-termc 49220

This theorem is referenced by: diagffth 49284

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