Theorem upciclem3 49329

Description: Lemma for upciclem4 49330. (Contributed by Zhi Wang, 17-Sep-2025.)

Hypotheses

Ref	Expression
upcic.b	⊢ 𝐵 = (Base‘𝐷)
upcic.c	⊢ 𝐶 = (Base‘𝐸)
upcic.h	⊢ 𝐻 = (Hom ‘𝐷)
upcic.j	⊢ 𝐽 = (Hom ‘𝐸)
upcic.o	⊢ 𝑂 = (comp‘𝐸)
upcic.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
upcic.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
upcic.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
upcic.z	⊢ (𝜑 → 𝑍 ∈ 𝐶)
upcic.m	⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
upcic.1	⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
upciclem3.od	⊢ · = (comp‘𝐷)
upciclem3.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
upciclem3.l	⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑋))
upciclem3.mn	⊢ (𝜑 → 𝑀 = (((𝑌𝐺𝑋)‘𝐿)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
upciclem3.nm	⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))

Assertion

Ref	Expression
upciclem3	⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) = ((Id‘𝐷)‘𝑋))

Distinct variable groups:   𝑤,𝐵   𝑓,𝐹,𝑘,𝑤   𝑓,𝐺,𝑘,𝑤   𝑓,𝐻,𝑘,𝑤   𝑓,𝐽,𝑤   𝑓,𝑀,𝑘,𝑤   𝑓,𝑂,𝑘,𝑤   𝑓,𝑋,𝑘,𝑤   𝑓,𝑍,𝑘,𝑤

Allowed substitution hints:   𝜑(𝑤,𝑓,𝑘)   𝐵(𝑓,𝑘)   𝐶(𝑤,𝑓,𝑘)   𝐷(𝑤,𝑓,𝑘)   · (𝑤,𝑓,𝑘)   𝐸(𝑤,𝑓,𝑘)   𝐽(𝑘)   𝐾(𝑤,𝑓,𝑘)   𝐿(𝑤,𝑓,𝑘)   𝑁(𝑤,𝑓,𝑘)   𝑌(𝑤,𝑓,𝑘)

Proof of Theorem upciclem3

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6831	. . . 4 ⊢ (𝑝 = (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾)))
2	1	oveq1d 7370	. . 3 ⊢ (𝑝 = (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) → (((𝑋𝐺𝑋)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
3	2	eqeq2d 2744	. 2 ⊢ (𝑝 = (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) → (𝑀 = (((𝑋𝐺𝑋)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) ↔ 𝑀 = (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀)))
4		fveq2 6831	. . . 4 ⊢ (𝑝 = ((Id‘𝐷)‘𝑋) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
5	4	oveq1d 7370	. . 3 ⊢ (𝑝 = ((Id‘𝐷)‘𝑋) → (((𝑋𝐺𝑋)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
6	5	eqeq2d 2744	. 2 ⊢ (𝑝 = ((Id‘𝐷)‘𝑋) → (𝑀 = (((𝑋𝐺𝑋)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) ↔ 𝑀 = (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀)))
7		upcic.1	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ 𝐵 ∀𝑓 ∈ (𝑍𝐽(𝐹‘𝑤))∃!𝑘 ∈ (𝑋𝐻𝑤)𝑓 = (((𝑋𝐺𝑤)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑤))𝑀))
8		upcic.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		upcic.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑍𝐽(𝐹‘𝑋)))
10	7, 8, 9	upciclem1 49327	. 2 ⊢ (𝜑 → ∃!𝑝 ∈ (𝑋𝐻𝑋)𝑀 = (((𝑋𝐺𝑋)‘𝑝)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
11		upcic.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
12		upcic.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
13		upciclem3.od	. . 3 ⊢ · = (comp‘𝐷)
14		upcic.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
15	14	funcrcl2 49240	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
16		upcic.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17		upciclem3.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
18		upciclem3.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑋))
19	11, 12, 13, 15, 8, 16, 8, 17, 18	catcocl 17599	. 2 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑋𝐻𝑋))
20		eqid 2733	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
21	11, 12, 20, 15, 8	catidcl 17596	. 2 ⊢ (𝜑 → ((Id‘𝐷)‘𝑋) ∈ (𝑋𝐻𝑋))
22		upciclem3.mn	. . 3 ⊢ (𝜑 → 𝑀 = (((𝑌𝐺𝑋)‘𝐿)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
23		upcic.c	. . . 4 ⊢ 𝐶 = (Base‘𝐸)
24		upcic.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
25		upcic.o	. . . 4 ⊢ 𝑂 = (comp‘𝐸)
26		upcic.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐶)
27		upciclem3.nm	. . . 4 ⊢ (𝜑 → 𝑁 = (((𝑋𝐺𝑌)‘𝐾)(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑌))𝑀))
28	11, 23, 12, 24, 25, 14, 8, 16, 8, 26, 9, 13, 17, 18, 27	upciclem2 49328	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = (((𝑌𝐺𝑋)‘𝐿)(⟨𝑍, (𝐹‘𝑌)⟩𝑂(𝐹‘𝑋))𝑁))
29	22, 28	eqtr4d 2771	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
30		eqid 2733	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
31	11, 20, 30, 14, 8	funcid 17785	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
32	31	oveq1d 7370	. . 3 ⊢ (𝜑 → (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = (((Id‘𝐸)‘(𝐹‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
33	14	funcrcl3 49241	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
34	11, 23, 14	funcf1 17781	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
35	34, 8	ffvelcdmd 7027	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
36	23, 24, 30, 33, 26, 25, 35, 9	catlid 17597	. . 3 ⊢ (𝜑 → (((Id‘𝐸)‘(𝐹‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = 𝑀)
37	32, 36	eqtr2d 2769	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
38	3, 6, 10, 19, 21, 29, 37	reu2eqd 3691	1 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) = ((Id‘𝐷)‘𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃!wreu 3345  ⟨cop 4583   class class class wbr 5095  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  Hom chom 17179  compcco 17180  Idccid 17579   Func cfunc 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-map 8761  df-ixp 8832  df-cat 17582  df-cid 17583  df-func 17773

This theorem is referenced by:  upciclem4  49330