Theorem upciclem3 49979

Description: Lemma for upciclem4 49980. (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 6885	. . . 4 ⊢ (𝑝 = (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾)))
2	1	oveq1d 7431	. . 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 2776	. 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 6885	. . . 4 ⊢ (𝑝 = ((Id‘𝐷)‘𝑋) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
5	4	oveq1d 7431	. . 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 2776	. 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 49977	. 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 49890	. . 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 17759	. 2 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑋𝐻𝑋))
20		eqid 2765	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
21	11, 12, 20, 15, 8	catidcl 17756	. 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 49978	. . 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 2803	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
30		eqid 2765	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
31	11, 20, 30, 14, 8	funcid 17945	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
32	31	oveq1d 7431	. . 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 49891	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
34	11, 23, 14	funcf1 17941	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
35	34, 8	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
36	23, 24, 30, 33, 26, 25, 35, 9	catlid 17757	. . 3 ⊢ (𝜑 → (((Id‘𝐸)‘(𝐹‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = 𝑀)
37	32, 36	eqtr2d 2801	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
38	3, 6, 10, 19, 21, 29, 37	reu2eqd 3701	1 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) = ((Id‘𝐷)‘𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃!wreu 3369  ⟨cop 4597   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  Hom chom 17339  compcco 17340  Idccid 17739   Func cfunc 17929

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-map 8828  df-ixp 8898  df-cat 17742  df-cid 17743  df-func 17933

This theorem is used by:  upciclem4  49980