Theorem upciclem3 49658

Description: Lemma for upciclem4 49659. (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 6827	. . . 4 ⊢ (𝑝 = (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾)))
2	1	oveq1d 7371	. . 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 2750	. 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 6827	. . . 4 ⊢ (𝑝 = ((Id‘𝐷)‘𝑋) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
5	4	oveq1d 7371	. . 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 2750	. 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 49656	. 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 49569	. . 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 17642	. 2 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑋𝐻𝑋))
20		eqid 2739	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
21	11, 12, 20, 15, 8	catidcl 17639	. 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 49657	. . 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 2777	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
30		eqid 2739	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
31	11, 20, 30, 14, 8	funcid 17828	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
32	31	oveq1d 7371	. . 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 49570	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
34	11, 23, 14	funcf1 17824	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
35	34, 8	ffvelcdmd 7026	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
36	23, 24, 30, 33, 26, 25, 35, 9	catlid 17640	. . 3 ⊢ (𝜑 → (((Id‘𝐸)‘(𝐹‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = 𝑀)
37	32, 36	eqtr2d 2775	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
38	3, 6, 10, 19, 21, 29, 37	reu2eqd 3677	1 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) = ((Id‘𝐷)‘𝑋))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃!wreu 3342  ⟨cop 4561   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  Hom chom 17222  compcco 17223  Idccid 17622   Func cfunc 17812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-ixp 8836  df-cat 17625  df-cid 17626  df-func 17816

This theorem is referenced by:  upciclem4  49659