Theorem diag2f1olem 49520

Description: Lemma for diag2f1o 49521. (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 17898	. . . . 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 17900	. . . 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 49208	. . . . . . 7 ⊢ (𝜑 → (1st ‘((1st ‘𝐿)‘𝑋))(𝐷 Func 𝐶)(2nd ‘((1st ‘𝐿)‘𝑋)))
11	10	funcrcl3 49064	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12		diag2f1o.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ TermCat)
13	12	termccd 49463	. . . . . 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 18186	. . . . 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 18186	. . . . 5 ⊢ (𝜑 → ((1st ‘((1st ‘𝐿)‘𝑌))‘𝑍) = 𝑌)
21	17, 20	oveq12d 7388	. . . 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 49519	. . 3 ⊢ (𝜑 → 𝑀 = {⟨𝑍, 𝐹⟩})
25	9, 14, 5, 6, 11, 13, 15, 18, 23	diag2 18188	. . . 4 ⊢ (𝜑 → ((𝑋(2nd ‘𝐿)𝑌)‘𝐹) = (𝐵 × {𝐹}))
26	12, 5, 7	termcbas2 49466	. . . . 5 ⊢ (𝜑 → 𝐵 = {𝑍})
27	26	xpeq1d 5660	. . . 4 ⊢ (𝜑 → (𝐵 × {𝐹}) = ({𝑍} × {𝐹}))
28		xpsng 7094	. . . . 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 4585  ⟨cop 4591   × cxp 5629  ‘cfv 6500  (class class class)co 7370  1^st c1st 7946  2^nd c2nd 7947  Basecbs 17157  Hom chom 17209   Nat cnat 17888  Δ_funccdiag 18155  TermCatctermc 49456

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7692  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6453  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7824  df-1st 7948  df-2nd 7949  df-frecs 8238  df-wrecs 8269  df-recs 8318  df-rdg 8356  df-1o 8412  df-er 8649  df-map 8779  df-ixp 8849  df-en 8897  df-dom 8898  df-sdom 8899  df-fin 8900  df-pnf 11189  df-mnf 11190  df-xr 11191  df-ltxr 11192  df-le 11193  df-sub 11386  df-neg 11387  df-nn 12166  df-2 12228  df-3 12229  df-4 12230  df-5 12231  df-6 12232  df-7 12233  df-8 12234  df-9 12235  df-n0 12422  df-z 12509  df-dec 12629  df-uz 12773  df-fz 13448  df-struct 17095  df-slot 17130  df-ndx 17142  df-base 17158  df-hom 17222  df-cco 17223  df-cat 17611  df-cid 17612  df-func 17802  df-nat 17890  df-xpc 18115  df-1stf 18116  df-curf 18157  df-diag 18159  df-thinc 49402  df-termc 49457

This theorem is referenced by:  diag2f1o  49521