Theorem diag2f1olem 50879

Description: Lemma for diag2f1o 50880. (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 18158	. . . . 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 18160	. . . 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 50545	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋))(𝐷 Func 𝐶)(2nd ‘((1st ‘𝐿)‘𝑋)))
11	10	funcrcl3 50399	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		diag2f1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ TermCat)
13	12	termccatd 50819	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		diag2f1o.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
15		diag2f1o.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		eqid 2763	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
17	9, 11, 13, 14, 15, 16, 5, 7	diag11 18446	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍) = 𝑋)
18		diag2f1o.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
19		eqid 2763	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑌) = ((1st ‘𝐿)‘𝑌)
20	9, 11, 13, 14, 18, 19, 5, 7	diag11 18446	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍) = 𝑌)
21	17, 20	oveq12d 7461	. . . 4 ⊢ (𝜑 → (((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍)𝐻((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍)) = (𝑋𝐻𝑌))
22	8, 21	eleqtrd 2866	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ (𝑋𝐻𝑌))
23	1, 22	eqeltrid 2868	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
24	12, 2, 3, 5, 7, 1	termcnatval 50878	. . 3 ⊢ (𝜑 → 𝑀 = {⟨𝑍, 𝐹⟩})
25	9, 14, 5, 6, 11, 13, 15, 18, 23	diag2 18448	. . . 4 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
26	12, 5, 7	termcbas2 50822	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑍})
27	26	xpeq1d 5684	. . . 4 ⊢ (𝜑 → (𝐵 × {𝐹}) = ({𝑍} × {𝐹}))
28		xpsng 7164	. . . . 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 2802	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = {⟨𝑍, 𝐹⟩})
31	24, 30	eqtr4d 2801	. 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 2147  {csn 4588  ⟨cop 4838   × cxp 5653  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Basecbs 17416  Hom chom 17468   Nat cnat 18148  Δ_funccdiag 18415  TermCatctermc 50812

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-struct 17354  df-slot 17389  df-ndx 17401  df-base 17417  df-hom 17481  df-cco 17482  df-cat 17871  df-cid 17872  df-func 18062  df-nat 18150  df-xpc 18375  df-1stf 18376  df-curf 18417  df-diag 18419  df-thinc 50758  df-termc 50813

This theorem is used by:  diag2f1o  50880