Theorem diag2f1olem 49522

Description: Lemma for diag2f1o 49523. (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 17916	. . . . 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 17918	. . . 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 49210	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋))(𝐷 Func 𝐶)(2nd ‘((1st ‘𝐿)‘𝑋)))
11	10	funcrcl3 49066	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		diag2f1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ TermCat)
13	12	termccd 49465	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		diag2f1o.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝐶)
15		diag2f1o.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16		eqid 2729	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑋) = ((1st ‘𝐿)‘𝑋)
17	9, 11, 13, 14, 15, 16, 5, 7	diag11 18204	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍) = 𝑋)
18		diag2f1o.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
19		eqid 2729	. . . . . 6 ⊢ ((1st ‘𝐿)‘𝑌) = ((1st ‘𝐿)‘𝑌)
20	9, 11, 13, 14, 18, 19, 5, 7	diag11 18204	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍) = 𝑌)
21	17, 20	oveq12d 7405	. . . 4 ⊢ (𝜑 → (((1st ‘((1st ‘𝐿)‘𝑋))‘𝑍)𝐻((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍)) = (𝑋𝐻𝑌))
22	8, 21	eleqtrd 2830	. . 3 ⊢ (𝜑 → (𝑀‘𝑍) ∈ (𝑋𝐻𝑌))
23	1, 22	eqeltrid 2832	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
24	12, 2, 3, 5, 7, 1	termcnatval 49521	. . 3 ⊢ (𝜑 → 𝑀 = {⟨𝑍, 𝐹⟩})
25	9, 14, 5, 6, 11, 13, 15, 18, 23	diag2 18206	. . . 4 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
26	12, 5, 7	termcbas2 49468	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑍})
27	26	xpeq1d 5667	. . . 4 ⊢ (𝜑 → (𝐵 × {𝐹}) = ({𝑍} × {𝐹}))
28		xpsng 7111	. . . . 5 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → ({𝑍} × {𝐹}) = {⟨𝑍, 𝐹⟩})
29	7, 23, 28	syl2anc 584	. . . 4 ⊢ (𝜑 → ({𝑍} × {𝐹}) = {⟨𝑍, 𝐹⟩})
30	25, 27, 29	3eqtrd 2768	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = {⟨𝑍, 𝐹⟩})
31	24, 30	eqtr4d 2767	. 2 ⊢ (𝜑 → 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹))
32	23, 31	jca 511	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝑀 = ((𝑋(2nd ‘𝐿)𝑌)‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4589  ⟨cop 4595   × cxp 5636  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231   Nat cnat 17906  Δ_funccdiag 18173  TermCatctermc 49458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-nat 17908  df-xpc 18133  df-1stf 18134  df-curf 18175  df-diag 18177  df-thinc 49404  df-termc 49459

This theorem is referenced by:  diag2f1o  49523