Theorem diag2f1olem 50566

Description: Lemma for diag2f1o 50567. (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 18090	. . . . 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 18092	. . . 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 50254	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋))(𝐷 Func 𝐶)(2nd ‘((1st ‘𝐿)‘𝑋)))
11	10	funcrcl3 50110	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		diag2f1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ TermCat)
13	12	termccatd 50509	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		diag2f1o.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
15		diag2f1o.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		eqid 2760	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
17	9, 11, 13, 14, 15, 16, 5, 7	diag11 18378	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍) = 𝑋)
18		diag2f1o.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
19		eqid 2760	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑌) = ((1st ‘𝐿)‘𝑌)
20	9, 11, 13, 14, 18, 19, 5, 7	diag11 18378	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍) = 𝑌)
21	17, 20	oveq12d 7426	. . . 4 ⊢ (𝜑 → (((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍)𝐻((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍)) = (𝑋𝐻𝑌))
22	8, 21	eleqtrd 2862	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ (𝑋𝐻𝑌))
23	1, 22	eqeltrid 2864	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
24	12, 2, 3, 5, 7, 1	termcnatval 50565	. . 3 ⊢ (𝜑 → 𝑀 = {⟨𝑍, 𝐹⟩})
25	9, 14, 5, 6, 11, 13, 15, 18, 23	diag2 18380	. . . 4 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
26	12, 5, 7	termcbas2 50512	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑍})
27	26	xpeq1d 5676	. . . 4 ⊢ (𝜑 → (𝐵 × {𝐹}) = ({𝑍} × {𝐹}))
28		xpsng 7128	. . . . 5 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → ({𝑍} × {𝐹}) = {⟨𝑍, 𝐹⟩})
29	7, 23, 28	syl2anc 596	. . . 4 ⊢ (𝜑 → ({𝑍} × {𝐹}) = {⟨𝑍, 𝐹⟩})
30	25, 27, 29	3eqtrd 2799	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = {⟨𝑍, 𝐹⟩})
31	24, 30	eqtr4d 2798	. 2 ⊢ (𝜑 → 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹))
32	23, 31	jca 521	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4583  ⟨cop 4589   × cxp 5645  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17348  Hom chom 17400   Nat cnat 18080  Δ_funccdiag 18347  TermCatctermc 50502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-struct 17286  df-slot 17321  df-ndx 17333  df-base 17349  df-hom 17413  df-cco 17414  df-cat 17803  df-cid 17804  df-func 17994  df-nat 18082  df-xpc 18307  df-1stf 18308  df-curf 18349  df-diag 18351  df-thinc 50448  df-termc 50503

This theorem is used by:  diag2f1o  50567