Theorem diag2f1olem 50342

Description: Lemma for diag2f1o 50343. (Contributed by Zhi Wang, 21-Oct-2025.)

Hypotheses

Ref	Expression
diag2f1o.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
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)
diag2f1olem.m	⊢ (𝜑 → 𝑀 ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
diag2f1olem.b	⊢ 𝐵 = (Base‘𝐷)
diag2f1olem.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
diag2f1olem.f	⊢ 𝐹 = (𝑀‘𝑍)

Assertion

Ref	Expression
diag2f1olem	⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹)))

Proof of Theorem diag2f1olem

Step	Hyp	Ref	Expression
1		diag2f1olem.f	. . 3 ⊢ 𝐹 = (𝑀‘𝑍)
2		diag2f1o.n	. . . . 5 ⊢ 𝑁 = (𝐷 Nat 𝐶)
3		diag2f1olem.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (((1st ‘𝐿)‘𝑋)𝑁((1st ‘𝐿)‘𝑌)))
4	2, 3	nat1st2nd 18017	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (⟨(1st ‘((1st ‘𝐿)‘𝑋)), (2nd ‘((1st ‘𝐿)‘𝑋))⟩𝑁⟨(1st ‘((1st ‘𝐿)‘𝑌)), (2nd ‘((1st ‘𝐿)‘𝑌))⟩))
5		diag2f1olem.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐷)
6		diag2f1o.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
7		diag2f1olem.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8	2, 4, 5, 6, 7	natcl 18019	. . . 4 ⊢ (𝜑 → (𝑀‘𝑍) ∈ (((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍)𝐻((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍)))
9		diag2f1o.l	. . . . . 6 ⊢ 𝐿 = (𝐶Δfunc𝐷)
10	2, 4	natrcl2 50030	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋))(𝐷 Func 𝐶)(2nd ‘((1st ‘𝐿)‘𝑋)))
11	10	funcrcl3 49886	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		diag2f1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ TermCat)
13	12	termccd 50285	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		diag2f1o.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
15		diag2f1o.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		eqid 2762	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
17	9, 11, 13, 14, 15, 16, 5, 7	diag11 18305	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍) = 𝑋)
18		diag2f1o.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
19		eqid 2762	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑌) = ((1st ‘𝐿)‘𝑌)
20	9, 11, 13, 14, 18, 19, 5, 7	diag11 18305	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍) = 𝑌)
21	17, 20	oveq12d 7430	. . . 4 ⊢ (𝜑 → (((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍)𝐻((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍)) = (𝑋𝐻𝑌))
22	8, 21	eleqtrd 2864	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ (𝑋𝐻𝑌))
23	1, 22	eqeltrid 2866	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
24	12, 2, 3, 5, 7, 1	termcnatval 50341	. . 3 ⊢ (𝜑 → 𝑀 = {⟨𝑍, 𝐹⟩})
25	9, 14, 5, 6, 11, 13, 15, 18, 23	diag2 18307	. . . 4 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
26	12, 5, 7	termcbas2 50288	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑍})
27	26	xpeq1d 5689	. . . 4 ⊢ (𝜑 → (𝐵 × {𝐹}) = ({𝑍} × {𝐹}))
28		xpsng 7135	. . . . 5 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → ({𝑍} × {𝐹}) = {⟨𝑍, 𝐹⟩})
29	7, 23, 28	syl2anc 595	. . . 4 ⊢ (𝜑 → ({𝑍} × {𝐹}) = {⟨𝑍, 𝐹⟩})
30	25, 27, 29	3eqtrd 2801	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = {⟨𝑍, 𝐹⟩})
31	24, 30	eqtr4d 2800	. 2 ⊢ (𝜑 → 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹))
32	23, 31	jca 520	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {csn 4588  ⟨cop 4594   × cxp 5658  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17275  Hom chom 17327   Nat cnat 18007  Δ_funccdiag 18274  TermCatctermc 50278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-struct 17213  df-slot 17248  df-ndx 17260  df-base 17276  df-hom 17340  df-cco 17341  df-cat 17730  df-cid 17731  df-func 17921  df-nat 18009  df-xpc 18234  df-1stf 18235  df-curf 18276  df-diag 18278  df-thinc 50224  df-termc 50279

This theorem is used by:  diag2f1o  50343