Theorem upciclem3 49750

Description: Lemma for upciclem4 49751. (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 6862	. . . 4 ⊢ (𝑝 = (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾)))
2	1	oveq1d 7406	. . 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 2772	. 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 6862	. . . 4 ⊢ (𝑝 = ((Id‘𝐷)‘𝑋) → ((𝑋𝐺𝑋)‘𝑝) = ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)))
5	4	oveq1d 7406	. . 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 2772	. 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 49748	. 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 49661	. . 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 17708	. 2 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) ∈ (𝑋𝐻𝑋))
20		eqid 2761	. . 3 ⊢ (Id‘𝐷) = (Id‘𝐷)
21	11, 12, 20, 15, 8	catidcl 17705	. 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 49749	. . 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 2799	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘(𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
30		eqid 2761	. . . . 5 ⊢ (Id‘𝐸) = (Id‘𝐸)
31	11, 20, 30, 14, 8	funcid 17894	. . . 4 ⊢ (𝜑 → ((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋)) = ((Id‘𝐸)‘(𝐹‘𝑋)))
32	31	oveq1d 7406	. . 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 49662	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
34	11, 23, 14	funcf1 17890	. . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
35	34, 8	ffvelcdmd 7061	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐶)
36	23, 24, 30, 33, 26, 25, 35, 9	catlid 17706	. . 3 ⊢ (𝜑 → (((Id‘𝐸)‘(𝐹‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀) = 𝑀)
37	32, 36	eqtr2d 2797	. 2 ⊢ (𝜑 → 𝑀 = (((𝑋𝐺𝑋)‘((Id‘𝐷)‘𝑋))(⟨𝑍, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑋))𝑀))
38	3, 6, 10, 19, 21, 29, 37	reu2eqd 3697	1 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩ · 𝑋)𝐾) = ((Id‘𝐷)‘𝑋))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃!wreu 3364  ⟨cop 4585   class class class wbr 5097  ‘cfv 6516  (class class class)co 7391  Basecbs 17236  Hom chom 17288  compcco 17289  Idccid 17688   Func cfunc 17878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-ixp 8874  df-cat 17691  df-cid 17692  df-func 17882

This theorem is referenced by:  upciclem4  49751