diag2f1o - Mathbox for Zhi Wang

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Theorem diag2f1o 50032

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 2737	. . 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 49974	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
8		diag2f1o.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
9		diag2f1o.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
10	3	istermc2 49970	. . . . . . 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 2578	. . . . 5 ⊢ (∃!𝑧 𝑧 ∈ (Base‘𝐷) → ∃𝑧 𝑧 ∈ (Base‘𝐷))
14	12, 13	syl 17	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧 ∈ (Base‘𝐷))
15		n0 4294	. . . 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 49804	. 2 ⊢ (𝜑 → (𝑋(2^nd ‘𝐿)𝑌):(𝑋𝐻𝑌)–1-1→(((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌)))
19		f1f 6734	. . . 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 49975	. . . . . 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 6838	. . . . . . 7 ⊢ (𝑓 = (𝑚‘𝑧) → ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓) = ((𝑋(2^nd ‘𝐿)𝑌)‘(𝑚‘𝑧)))
24	23	eqeq2d 2748	. . . . . 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 4608	. . . . . . . . 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 2844	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))) ∧ (Base‘𝐷) = {𝑧}) → 𝑧 ∈ (Base‘𝐷))
32		eqid 2737	. . . . . . . 8 ⊢ (𝑚‘𝑧) = (𝑚‘𝑧)
33	1, 2, 4, 25, 26, 17, 27, 28, 3, 31, 32	diag2f1olem 50031	. . . . . . 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 3568	. . . . 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 3130	. . 3 ⊢ (𝜑 → ∀𝑚 ∈ (((1^st ‘𝐿)‘𝑋)𝑁((1^st ‘𝐿)‘𝑌))∃𝑓 ∈ (𝑋𝐻𝑌)𝑚 = ((𝑋(2^nd ‘𝐿)𝑌)‘𝑓))
39		dffo3 7052	. . 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 6503	. 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 2569 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∅c0 4274 {csn 4568 ⟶wf 6492 –1-1→wf1 6493 –onto→wfo 6494 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7364 1^st c1st 7937 2^nd c2nd 7938 Basecbs 17176 Hom chom 17228 Catccat 17627 Nat cnat 17908 Δ_funccdiag 18175 ThinCatcthinc 49912 TermCatctermc 49967

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 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-2 12241 df-3 12242 df-4 12243 df-5 12244 df-6 12245 df-7 12246 df-8 12247 df-9 12248 df-n0 12435 df-z 12522 df-dec 12642 df-uz 12786 df-fz 13459 df-struct 17114 df-slot 17149 df-ndx 17161 df-base 17177 df-hom 17241 df-cco 17242 df-cat 17631 df-cid 17632 df-func 17822 df-nat 17910 df-fuc 17911 df-xpc 18135 df-1stf 18136 df-curf 18177 df-diag 18179 df-thinc 49913 df-termc 49968

This theorem is referenced by: diagffth 50033

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