uptr2 - Mathbox for Zhi Wang

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Theorem uptr2 49200

Description: Universal property and fully faithful functor surjective on objects. (Contributed by Zhi Wang, 25-Nov-2025.)

**Hypotheses**
Ref	Expression
uptr2.a	⊢ 𝐴 = (Base‘𝐶)
uptr2.b	⊢ 𝐵 = (Base‘𝐷)
uptr2.y	⊢ (𝜑 → 𝑌 = (𝑅‘𝑋))
uptr2.r	⊢ (𝜑 → 𝑅:𝐴–onto→𝐵)
uptr2.s	⊢ (𝜑 → 𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆)
uptr2.f	⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘_func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩)
uptr2.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
uptr2.k	⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)

**Assertion**
Ref	Expression
uptr2	⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀))

Proof of Theorem uptr2 Dummy variables 𝑔 𝑘 𝑙 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀) → 𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀)
2		eqid 2730	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
3	1, 2	uprcl3 49169	. . 3 ⊢ ((𝜑 ∧ 𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀) → 𝑍 ∈ (Base‘𝐸))
4		eqid 2730	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
5	1, 4	uprcl5 49171	. . 3 ⊢ ((𝜑 ∧ 𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀) → 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))
6	3, 5	jca 511	. 2 ⊢ ((𝜑 ∧ 𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀) → (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋))))
7		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀) → 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀)
8	7, 2	uprcl3 49169	. . 3 ⊢ ((𝜑 ∧ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀) → 𝑍 ∈ (Base‘𝐸))
9	7, 4	uprcl5 49171	. . . 4 ⊢ ((𝜑 ∧ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀) → 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐾‘𝑌)))
10		uptr2.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 = (𝑅‘𝑋))
11	10	fveq2d 6864	. . . . . . 7 ⊢ (𝜑 → (𝐾‘𝑌) = (𝐾‘(𝑅‘𝑋)))
12		uptr2.a	. . . . . . . 8 ⊢ 𝐴 = (Base‘𝐶)
13		uptr2.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆)
14		inss1 4202	. . . . . . . . . . 11 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Full 𝐷)
15		fullfunc 17876	. . . . . . . . . . 11 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
16	14, 15	sstri 3958	. . . . . . . . . 10 ⊢ ((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷)) ⊆ (𝐶 Func 𝐷)
17	16	ssbri 5154	. . . . . . . . 9 ⊢ (𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆 → 𝑅(𝐶 Func 𝐷)𝑆)
18	13, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅(𝐶 Func 𝐷)𝑆)
19		uptr2.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
20		uptr2.f	. . . . . . . 8 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘_func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩)
21		uptr2.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22	12, 18, 19, 20, 21	cofu1a 49073	. . . . . . 7 ⊢ (𝜑 → (𝐾‘(𝑅‘𝑋)) = (𝐹‘𝑋))
23	11, 22	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (𝐾‘𝑌) = (𝐹‘𝑋))
24	23	oveq2d 7405	. . . . 5 ⊢ (𝜑 → (𝑍(Hom ‘𝐸)(𝐾‘𝑌)) = (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))
25	24	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀) → (𝑍(Hom ‘𝐸)(𝐾‘𝑌)) = (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))
26	9, 25	eleqtrd 2831	. . 3 ⊢ ((𝜑 ∧ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀) → 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))
27	8, 26	jca 511	. 2 ⊢ ((𝜑 ∧ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀) → (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋))))
28		uptr2.r	. . . . . . 7 ⊢ (𝜑 → 𝑅:𝐴–onto→𝐵)
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑅:𝐴–onto→𝐵)
30		fof 6774	. . . . . 6 ⊢ (𝑅:𝐴–onto→𝐵 → 𝑅:𝐴⟶𝐵)
31	29, 30	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑅:𝐴⟶𝐵)
32	31	ffvelcdmda 7058	. . . 4 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴) → (𝑅‘𝑥) ∈ 𝐵)
33		foelrn 7081	. . . . 5 ⊢ ((𝑅:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝑅‘𝑥))
34	29, 33	sylan 580	. . . 4 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝑅‘𝑥))
35		simp3 1138	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝑦 = (𝑅‘𝑥))
36	35	fveq2d 6864	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝐾‘𝑦) = (𝐾‘(𝑅‘𝑥)))
37		simp1l 1198	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝜑)
38	37, 18	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝑅(𝐶 Func 𝐷)𝑆)
39	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝐾(𝐷 Func 𝐸)𝐿)
40	39	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝐾(𝐷 Func 𝐸)𝐿)
41	37, 20	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (⟨𝐾, 𝐿⟩ ∘_func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩)
42		simp2 1137	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝑥 ∈ 𝐴)
43	12, 38, 40, 41, 42	cofu1a 49073	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝐾‘(𝑅‘𝑥)) = (𝐹‘𝑥))
44	36, 43	eqtrd 2765	. . . . . 6 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝐾‘𝑦) = (𝐹‘𝑥))
45	44	oveq2d 7405	. . . . 5 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝑍(Hom ‘𝐸)(𝐾‘𝑦)) = (𝑍(Hom ‘𝐸)(𝐹‘𝑥)))
46		eqid 2730	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
47		eqid 2730	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
48	37, 13	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝑅((𝐶 Full 𝐷) ∩ (𝐶 Faith 𝐷))𝑆)
49	37, 21	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝑋 ∈ 𝐴)
50	12, 46, 47, 48, 49, 42	ffthf1o 17889	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→((𝑅‘𝑋)(Hom ‘𝐷)(𝑅‘𝑥)))
51	37, 10	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → 𝑌 = (𝑅‘𝑋))
52	51, 35	oveq12d 7407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝑌(Hom ‘𝐷)𝑦) = ((𝑅‘𝑋)(Hom ‘𝐷)(𝑅‘𝑥)))
53	52	f1oeq3d 6799	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → ((𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→(𝑌(Hom ‘𝐷)𝑦) ↔ (𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→((𝑅‘𝑋)(Hom ‘𝐷)(𝑅‘𝑥))))
54	50, 53	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→(𝑌(Hom ‘𝐷)𝑦))
55		f1of 6802	. . . . . . . 8 ⊢ ((𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→(𝑌(Hom ‘𝐷)𝑦) → (𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)⟶(𝑌(Hom ‘𝐷)𝑦))
56	54, 55	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)⟶(𝑌(Hom ‘𝐷)𝑦))
57	56	ffvelcdmda 7058	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ 𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)) → ((𝑋𝑆𝑥)‘𝑘) ∈ (𝑌(Hom ‘𝐷)𝑦))
58		f1ofveu 7383	. . . . . . . 8 ⊢ (((𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→(𝑌(Hom ‘𝐷)𝑦) ∧ 𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)) → ∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)((𝑋𝑆𝑥)‘𝑘) = 𝑙)
59		eqcom 2737	. . . . . . . . 9 ⊢ (((𝑋𝑆𝑥)‘𝑘) = 𝑙 ↔ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))
60	59	reubii 3365	. . . . . . . 8 ⊢ (∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)((𝑋𝑆𝑥)‘𝑘) = 𝑙 ↔ ∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑙 = ((𝑋𝑆𝑥)‘𝑘))
61	58, 60	sylib 218	. . . . . . 7 ⊢ (((𝑋𝑆𝑥):(𝑋(Hom ‘𝐶)𝑥)–1-1-onto→(𝑌(Hom ‘𝐷)𝑦) ∧ 𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)) → ∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑙 = ((𝑋𝑆𝑥)‘𝑘))
62	54, 61	sylan 580	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ 𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)) → ∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑙 = ((𝑋𝑆𝑥)‘𝑘))
63	37, 23	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (𝐾‘𝑌) = (𝐹‘𝑋))
64	63	opeq2d 4846	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → ⟨𝑍, (𝐾‘𝑌)⟩ = ⟨𝑍, (𝐹‘𝑋)⟩)
65	64, 44	oveq12d 7407	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦)) = (⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥)))
66	65	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → (⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦)) = (⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥)))
67	51	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑌 = (𝑅‘𝑋))
68		simpl3 1194	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑦 = (𝑅‘𝑥))
69	67, 68	oveq12d 7407	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → (𝑌𝐿𝑦) = ((𝑅‘𝑋)𝐿(𝑅‘𝑥)))
70		simprr 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑙 = ((𝑋𝑆𝑥)‘𝑘))
71	69, 70	fveq12d 6867	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → ((𝑌𝐿𝑦)‘𝑙) = (((𝑅‘𝑋)𝐿(𝑅‘𝑥))‘((𝑋𝑆𝑥)‘𝑘)))
72	38	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑅(𝐶 Func 𝐷)𝑆)
73	40	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝐾(𝐷 Func 𝐸)𝐿)
74	41	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → (⟨𝐾, 𝐿⟩ ∘_func ⟨𝑅, 𝑆⟩) = ⟨𝐹, 𝐺⟩)
75	49	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑋 ∈ 𝐴)
76	42	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑥 ∈ 𝐴)
77		simprl 770	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥))
78	12, 72, 73, 74, 75, 76, 46, 77	cofu2a 49074	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → (((𝑅‘𝑋)𝐿(𝑅‘𝑥))‘((𝑋𝑆𝑥)‘𝑘)) = ((𝑋𝐺𝑥)‘𝑘))
79	71, 78	eqtrd 2765	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → ((𝑌𝐿𝑦)‘𝑙) = ((𝑋𝐺𝑥)‘𝑘))
80		eqidd 2731	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → 𝑀 = 𝑀)
81	66, 79, 80	oveq123d 7410	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → (((𝑌𝐿𝑦)‘𝑙)(⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦))𝑀) = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀))
82	81	eqeq2d 2741	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) ∧ (𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥) ∧ 𝑙 = ((𝑋𝑆𝑥)‘𝑘))) → (𝑔 = (((𝑌𝐿𝑦)‘𝑙)(⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦))𝑀) ↔ 𝑔 = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀)))
83	57, 62, 82	reuxfr1dd 48785	. . . . 5 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (∃!𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)𝑔 = (((𝑌𝐿𝑦)‘𝑙)(⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦))𝑀) ↔ ∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑔 = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀)))
84	45, 83	raleqbidv 3321	. . . 4 ⊢ (((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝑅‘𝑥)) → (∀𝑔 ∈ (𝑍(Hom ‘𝐸)(𝐾‘𝑦))∃!𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)𝑔 = (((𝑌𝐿𝑦)‘𝑙)(⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦))𝑀) ↔ ∀𝑔 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑥))∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑔 = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀)))
85	32, 34, 84	ralxfrd2 5369	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑍(Hom ‘𝐸)(𝐾‘𝑦))∃!𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)𝑔 = (((𝑌𝐿𝑦)‘𝑙)(⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦))𝑀) ↔ ∀𝑥 ∈ 𝐴 ∀𝑔 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑥))∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑔 = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀)))
86		uptr2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
87		eqid 2730	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
88		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑍 ∈ (Base‘𝐸))
89	10	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑌 = (𝑅‘𝑋))
90	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑋 ∈ 𝐴)
91	31, 90	ffvelcdmd 7059	. . . . 5 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → (𝑅‘𝑋) ∈ 𝐵)
92	89, 91	eqeltrd 2829	. . . 4 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑌 ∈ 𝐵)
93		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))
94	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → (𝑍(Hom ‘𝐸)(𝐾‘𝑌)) = (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))
95	93, 94	eleqtrrd 2832	. . . 4 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐾‘𝑌)))
96	86, 2, 47, 4, 87, 88, 39, 92, 95	isup 49159	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → (𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑍(Hom ‘𝐸)(𝐾‘𝑦))∃!𝑙 ∈ (𝑌(Hom ‘𝐷)𝑦)𝑔 = (((𝑌𝐿𝑦)‘𝑙)(⟨𝑍, (𝐾‘𝑌)⟩(comp‘𝐸)(𝐾‘𝑦))𝑀)))
97	18, 19	cofucla 49075	. . . . . . 7 ⊢ (𝜑 → (⟨𝐾, 𝐿⟩ ∘_func ⟨𝑅, 𝑆⟩) ∈ (𝐶 Func 𝐸))
98	20, 97	eqeltrrd 2830	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
99		df-br 5110	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐸))
100	98, 99	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐸)𝐺)
101	100	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → 𝐹(𝐶 Func 𝐸)𝐺)
102	12, 2, 46, 4, 87, 88, 101, 90, 93	isup 49159	. . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ ∀𝑥 ∈ 𝐴 ∀𝑔 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑥))∃!𝑘 ∈ (𝑋(Hom ‘𝐶)𝑥)𝑔 = (((𝑋𝐺𝑥)‘𝑘)(⟨𝑍, (𝐹‘𝑋)⟩(comp‘𝐸)(𝐹‘𝑥))𝑀)))
103	85, 96, 102	3bitr4rd 312	. 2 ⊢ ((𝜑 ∧ (𝑍 ∈ (Base‘𝐸) ∧ 𝑀 ∈ (𝑍(Hom ‘𝐸)(𝐹‘𝑋)))) → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀))
104	6, 27, 103	bibiad 839	1 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐶 UP 𝐸)𝑍)𝑀 ↔ 𝑌(⟨𝐾, 𝐿⟩(𝐷 UP 𝐸)𝑍)𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 ∃!wreu 3354 ∩ cin 3915 ⟨cop 4597 class class class wbr 5109 ⟶wf 6509 –onto→wfo 6511 –1-1-onto→wf1o 6512 ‘cfv 6513 (class class class)co 7389 Basecbs 17185 Hom chom 17237 compcco 17238 Func cfunc 17822 ∘_func ccofu 17824 Full cful 17872 Faith cfth 17873 UP cup 49152

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 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-1st 7970 df-2nd 7971 df-map 8803 df-ixp 8873 df-cat 17635 df-cid 17636 df-func 17826 df-cofu 17828 df-full 17874 df-fth 17875 df-up 49153

This theorem is referenced by: uptr2a 49201

W3C validator