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Theorem catcisolem 17023
Description: Lemma for catciso 17024. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
catciso.c 𝐶 = (CatCat‘𝑈)
catciso.b 𝐵 = (Base‘𝐶)
catciso.r 𝑅 = (Base‘𝑋)
catciso.s 𝑆 = (Base‘𝑌)
catciso.u (𝜑𝑈𝑉)
catciso.x (𝜑𝑋𝐵)
catciso.y (𝜑𝑌𝐵)
catcisolem.i 𝐼 = (Inv‘𝐶)
catcisolem.g 𝐻 = (𝑥𝑆, 𝑦𝑆((𝐹𝑥)𝐺(𝐹𝑦)))
catcisolem.1 (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
catcisolem.2 (𝜑𝐹:𝑅1-1-onto𝑆)
Assertion
Ref Expression
catcisolem (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)
Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem catcisolem
Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcisolem.2 . . . . . . 7 (𝜑𝐹:𝑅1-1-onto𝑆)
2 f1ococnv1 6348 . . . . . . 7 (𝐹:𝑅1-1-onto𝑆 → (𝐹𝐹) = ( I ↾ 𝑅))
31, 2syl 17 . . . . . 6 (𝜑 → (𝐹𝐹) = ( I ↾ 𝑅))
413ad2ant1 1163 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹:𝑅1-1-onto𝑆)
5 f1of 6320 . . . . . . . . . . . . . 14 (𝐹:𝑅1-1-onto𝑆𝐹:𝑅𝑆)
64, 5syl 17 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹:𝑅𝑆)
7 simp2 1167 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝑢𝑅)
86, 7ffvelrnd 6550 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹𝑢) ∈ 𝑆)
9 simp3 1168 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝑣𝑅)
106, 9ffvelrnd 6550 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹𝑣) ∈ 𝑆)
11 simpl 474 . . . . . . . . . . . . . . . 16 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → 𝑥 = (𝐹𝑢))
1211fveq2d 6379 . . . . . . . . . . . . . . 15 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → (𝐹𝑥) = (𝐹‘(𝐹𝑢)))
13 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → 𝑦 = (𝐹𝑣))
1413fveq2d 6379 . . . . . . . . . . . . . . 15 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → (𝐹𝑦) = (𝐹‘(𝐹𝑣)))
1512, 14oveq12d 6860 . . . . . . . . . . . . . 14 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
1615cnveqd 5466 . . . . . . . . . . . . 13 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
17 catcisolem.g . . . . . . . . . . . . 13 𝐻 = (𝑥𝑆, 𝑦𝑆((𝐹𝑥)𝐺(𝐹𝑦)))
18 ovex 6874 . . . . . . . . . . . . . 14 ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) ∈ V
1918cnvex 7311 . . . . . . . . . . . . 13 ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) ∈ V
2016, 17, 19ovmpt2a 6989 . . . . . . . . . . . 12 (((𝐹𝑢) ∈ 𝑆 ∧ (𝐹𝑣) ∈ 𝑆) → ((𝐹𝑢)𝐻(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
218, 10, 20syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹𝑢)𝐻(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
22 f1ocnvfv1 6724 . . . . . . . . . . . . . 14 ((𝐹:𝑅1-1-onto𝑆𝑢𝑅) → (𝐹‘(𝐹𝑢)) = 𝑢)
234, 7, 22syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹‘(𝐹𝑢)) = 𝑢)
24 f1ocnvfv1 6724 . . . . . . . . . . . . . 14 ((𝐹:𝑅1-1-onto𝑆𝑣𝑅) → (𝐹‘(𝐹𝑣)) = 𝑣)
254, 9, 24syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹‘(𝐹𝑣)) = 𝑣)
2623, 25oveq12d 6860 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) = (𝑢𝐺𝑣))
2726cnveqd 5466 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) = (𝑢𝐺𝑣))
2821, 27eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹𝑢)𝐻(𝐹𝑣)) = (𝑢𝐺𝑣))
2928coeq1d 5452 . . . . . . . . 9 ((𝜑𝑢𝑅𝑣𝑅) → (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)) = ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)))
30 catciso.r . . . . . . . . . . 11 𝑅 = (Base‘𝑋)
31 eqid 2765 . . . . . . . . . . 11 (Hom ‘𝑋) = (Hom ‘𝑋)
32 eqid 2765 . . . . . . . . . . 11 (Hom ‘𝑌) = (Hom ‘𝑌)
33 catcisolem.1 . . . . . . . . . . . 12 (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
34333ad2ant1 1163 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
3530, 31, 32, 34, 7, 9ffthf1o 16846 . . . . . . . . . 10 ((𝜑𝑢𝑅𝑣𝑅) → (𝑢𝐺𝑣):(𝑢(Hom ‘𝑋)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑌)(𝐹𝑣)))
36 f1ococnv1 6348 . . . . . . . . . 10 ((𝑢𝐺𝑣):(𝑢(Hom ‘𝑋)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑌)(𝐹𝑣)) → ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3735, 36syl 17 . . . . . . . . 9 ((𝜑𝑢𝑅𝑣𝑅) → ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3829, 37eqtrd 2799 . . . . . . . 8 ((𝜑𝑢𝑅𝑣𝑅) → (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3938mpt2eq3dva 6917 . . . . . . 7 (𝜑 → (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣))) = (𝑢𝑅, 𝑣𝑅 ↦ ( I ↾ (𝑢(Hom ‘𝑋)𝑣))))
40 fveq2 6375 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑋)‘𝑧) = ((Hom ‘𝑋)‘⟨𝑢, 𝑣⟩))
41 df-ov 6845 . . . . . . . . . 10 (𝑢(Hom ‘𝑋)𝑣) = ((Hom ‘𝑋)‘⟨𝑢, 𝑣⟩)
4240, 41syl6eqr 2817 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑋)‘𝑧) = (𝑢(Hom ‘𝑋)𝑣))
4342reseq2d 5565 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → ( I ↾ ((Hom ‘𝑋)‘𝑧)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
4443mpt2mpt 6950 . . . . . . 7 (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧))) = (𝑢𝑅, 𝑣𝑅 ↦ ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
4539, 44syl6eqr 2817 . . . . . 6 (𝜑 → (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣))) = (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧))))
463, 45opeq12d 4567 . . . . 5 (𝜑 → ⟨(𝐹𝐹), (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)))⟩ = ⟨( I ↾ 𝑅), (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧)))⟩)
47 inss1 3992 . . . . . . . . 9 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
48 fullfunc 16833 . . . . . . . . 9 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
4947, 48sstri 3770 . . . . . . . 8 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
5049ssbri 4854 . . . . . . 7 (𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺𝐹(𝑋 Func 𝑌)𝐺)
5133, 50syl 17 . . . . . 6 (𝜑𝐹(𝑋 Func 𝑌)𝐺)
52 catciso.s . . . . . . 7 𝑆 = (Base‘𝑌)
53 eqid 2765 . . . . . . 7 (Id‘𝑌) = (Id‘𝑌)
54 eqid 2765 . . . . . . 7 (Id‘𝑋) = (Id‘𝑋)
55 eqid 2765 . . . . . . 7 (comp‘𝑌) = (comp‘𝑌)
56 eqid 2765 . . . . . . 7 (comp‘𝑋) = (comp‘𝑋)
57 catciso.c . . . . . . . . . 10 𝐶 = (CatCat‘𝑈)
58 catciso.b . . . . . . . . . 10 𝐵 = (Base‘𝐶)
59 catciso.u . . . . . . . . . 10 (𝜑𝑈𝑉)
6057, 58, 59catcbas 17014 . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Cat))
61 inss2 3993 . . . . . . . . 9 (𝑈 ∩ Cat) ⊆ Cat
6260, 61syl6eqss 3815 . . . . . . . 8 (𝜑𝐵 ⊆ Cat)
63 catciso.y . . . . . . . 8 (𝜑𝑌𝐵)
6462, 63sseldd 3762 . . . . . . 7 (𝜑𝑌 ∈ Cat)
65 catciso.x . . . . . . . 8 (𝜑𝑋𝐵)
6662, 65sseldd 3762 . . . . . . 7 (𝜑𝑋 ∈ Cat)
67 f1ocnv 6332 . . . . . . . 8 (𝐹:𝑅1-1-onto𝑆𝐹:𝑆1-1-onto𝑅)
68 f1of 6320 . . . . . . . 8 (𝐹:𝑆1-1-onto𝑅𝐹:𝑆𝑅)
691, 67, 683syl 18 . . . . . . 7 (𝜑𝐹:𝑆𝑅)
70 ovex 6874 . . . . . . . . . 10 ((𝐹𝑥)𝐺(𝐹𝑦)) ∈ V
7170cnvex 7311 . . . . . . . . 9 ((𝐹𝑥)𝐺(𝐹𝑦)) ∈ V
7217, 71fnmpt2i 7440 . . . . . . . 8 𝐻 Fn (𝑆 × 𝑆)
7372a1i 11 . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
7433adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
7569ffvelrnda 6549 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → (𝐹𝑢) ∈ 𝑅)
7675adantrr 708 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹𝑢) ∈ 𝑅)
7769ffvelrnda 6549 . . . . . . . . . . 11 ((𝜑𝑣𝑆) → (𝐹𝑣) ∈ 𝑅)
7877adantrl 707 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹𝑣) ∈ 𝑅)
7930, 31, 32, 74, 76, 78ffthf1o 16846 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
80 f1ocnv 6332 . . . . . . . . 9 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
81 f1of 6320 . . . . . . . . 9 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
8279, 80, 813syl 18 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
83 simpl 474 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
8483fveq2d 6379 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝐹𝑥) = (𝐹𝑢))
85 simpr 477 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
8685fveq2d 6379 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝐹𝑦) = (𝐹𝑣))
8784, 86oveq12d 6860 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑣)))
8887cnveqd 5466 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑣)))
89 ovex 6874 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑣)) ∈ V
9089cnvex 7311 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑣)) ∈ V
9188, 17, 90ovmpt2a 6989 . . . . . . . . . 10 ((𝑢𝑆𝑣𝑆) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
9291adantl 473 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
931adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝐹:𝑅1-1-onto𝑆)
94 simprl 787 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝑢𝑆)
95 f1ocnvfv2 6725 . . . . . . . . . . . 12 ((𝐹:𝑅1-1-onto𝑆𝑢𝑆) → (𝐹‘(𝐹𝑢)) = 𝑢)
9693, 94, 95syl2anc 579 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹‘(𝐹𝑢)) = 𝑢)
97 simprr 789 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝑣𝑆)
98 f1ocnvfv2 6725 . . . . . . . . . . . 12 ((𝐹:𝑅1-1-onto𝑆𝑣𝑆) → (𝐹‘(𝐹𝑣)) = 𝑣)
9993, 97, 98syl2anc 579 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹‘(𝐹𝑣)) = 𝑣)
10096, 99oveq12d 6860 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
101100eqcomd 2771 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢(Hom ‘𝑌)𝑣) = ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
10292, 101feq12d 6211 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝑢𝐻𝑣):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣)))⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))))
10382, 102mpbird 248 . . . . . . 7 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝐻𝑣):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
104 simpr 477 . . . . . . . . . 10 ((𝜑𝑢𝑆) → 𝑢𝑆)
105 simpl 474 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑥 = 𝑢)
106105fveq2d 6379 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝐹𝑥) = (𝐹𝑢))
107 simpr 477 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑦 = 𝑢)
108107fveq2d 6379 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝐹𝑦) = (𝐹𝑢))
109106, 108oveq12d 6860 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑢)))
110109cnveqd 5466 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑢)))
111 ovex 6874 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑢)) ∈ V
112111cnvex 7311 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑢)) ∈ V
113110, 17, 112ovmpt2a 6989 . . . . . . . . . 10 ((𝑢𝑆𝑢𝑆) → (𝑢𝐻𝑢) = ((𝐹𝑢)𝐺(𝐹𝑢)))
114104, 104, 113syl2anc 579 . . . . . . . . 9 ((𝜑𝑢𝑆) → (𝑢𝐻𝑢) = ((𝐹𝑢)𝐺(𝐹𝑢)))
115114fveq1d 6377 . . . . . . . 8 ((𝜑𝑢𝑆) → ((𝑢𝐻𝑢)‘((Id‘𝑌)‘𝑢)) = (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)))
11651adantr 472 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝐹(𝑋 Func 𝑌)𝐺)
11730, 54, 53, 116, 75funcid 16797 . . . . . . . . . 10 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘(𝐹‘(𝐹𝑢))))
1181, 95sylan 575 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → (𝐹‘(𝐹𝑢)) = 𝑢)
119118fveq2d 6379 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((Id‘𝑌)‘(𝐹‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢))
120117, 119eqtrd 2799 . . . . . . . . 9 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢))
12133adantr 472 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
12230, 31, 32, 121, 75, 75ffthf1o 16846 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((𝐹𝑢)𝐺(𝐹𝑢)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑢))))
12366adantr 472 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝑋 ∈ Cat)
12430, 31, 54, 123, 75catidcl 16610 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((Id‘𝑋)‘(𝐹𝑢)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢)))
125 f1ocnvfv 6726 . . . . . . . . . 10 ((((𝐹𝑢)𝐺(𝐹𝑢)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑢))) ∧ ((Id‘𝑋)‘(𝐹𝑢)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))) → ((((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢))))
126122, 124, 125syl2anc 579 . . . . . . . . 9 ((𝜑𝑢𝑆) → ((((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢))))
127120, 126mpd 15 . . . . . . . 8 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢)))
128115, 127eqtrd 2799 . . . . . . 7 ((𝜑𝑢𝑆) → ((𝑢𝐻𝑢)‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢)))
129513ad2ant1 1163 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹(𝑋 Func 𝑌)𝐺)
130693ad2ant1 1163 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹:𝑆𝑅)
131 simp21 1263 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑢𝑆)
132130, 131ffvelrnd 6550 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑢) ∈ 𝑅)
133 simp22 1264 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑣𝑆)
134130, 133ffvelrnd 6550 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑣) ∈ 𝑅)
135 simp23 1265 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑧𝑆)
136130, 135ffvelrnd 6550 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑧) ∈ 𝑅)
137333ad2ant1 1163 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
13830, 31, 32, 137, 132, 134ffthf1o 16846 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
13913ad2ant1 1163 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹:𝑅1-1-onto𝑆)
140139, 131, 95syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑢)) = 𝑢)
141139, 133, 98syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑣)) = 𝑣)
142140, 141oveq12d 6860 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
143 f1oeq3 6312 . . . . . . . . . . . . . . 15 (((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣) → (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣)))
144142, 143syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣)))
145138, 144mpbid 223 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣))
146 f1ocnv 6332 . . . . . . . . . . . . 13 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣) → ((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
147 f1of 6320 . . . . . . . . . . . . 13 (((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)) → ((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
148145, 146, 1473syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑣)):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
149 simp3l 1258 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣))
150148, 149ffvelrnd 6550 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
15130, 31, 32, 137, 134, 136ffthf1o 16846 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))))
152 f1ocnvfv2 6725 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑅1-1-onto𝑆𝑧𝑆) → (𝐹‘(𝐹𝑧)) = 𝑧)
153139, 135, 152syl2anc 579 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
154141, 153oveq12d 6860 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) = (𝑣(Hom ‘𝑌)𝑧))
155 f1oeq3 6312 . . . . . . . . . . . . . . 15 (((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) = (𝑣(Hom ‘𝑌)𝑧) → (((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) ↔ ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧)))
156154, 155syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑣))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) ↔ ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧)))
157151, 156mpbid 223 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧))
158 f1ocnv 6332 . . . . . . . . . . . . 13 (((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧) → ((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)–1-1-onto→((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
159 f1of 6320 . . . . . . . . . . . . 13 (((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)–1-1-onto→((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)) → ((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)⟶((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
160157, 158, 1593syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑣)𝐺(𝐹𝑧)):(𝑣(Hom ‘𝑌)𝑧)⟶((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
161 simp3r 1259 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))
162160, 161ffvelrnd 6550 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔) ∈ ((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧)))
16330, 31, 56, 55, 129, 132, 134, 136, 150, 162funcco 16798 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))(⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩(comp‘𝑌)(𝐹‘(𝐹𝑧)))(((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))))
164140, 141opeq12d 4567 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩ = ⟨𝑢, 𝑣⟩)
165164, 153oveq12d 6860 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩(comp‘𝑌)(𝐹‘(𝐹𝑧))) = (⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧))
166 f1ocnvfv2 6725 . . . . . . . . . . . 12 ((((𝐹𝑣)𝐺(𝐹𝑧)):((𝐹𝑣)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑣(Hom ‘𝑌)𝑧) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧)) → (((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)) = 𝑔)
167157, 161, 166syl2anc 579 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)) = 𝑔)
168 f1ocnvfv2 6725 . . . . . . . . . . . 12 ((((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣) ∧ 𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣)) → (((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) = 𝑓)
169145, 149, 168syl2anc 579 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) = 𝑓)
170165, 167, 169oveq123d 6863 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))(⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩(comp‘𝑌)(𝐹‘(𝐹𝑧)))(((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓))
171163, 170eqtrd 2799 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓))
17230, 31, 32, 137, 132, 136ffthf1o 16846 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))))
173140, 153oveq12d 6860 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) = (𝑢(Hom ‘𝑌)𝑧))
174 f1oeq3 6312 . . . . . . . . . . . 12 (((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) = (𝑢(Hom ‘𝑌)𝑧) → (((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) ↔ ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧)))
175173, 174syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑧))) ↔ ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧)))
176172, 175mpbid 223 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧))
177663ad2ant1 1163 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑋 ∈ Cat)
17830, 31, 56, 177, 132, 134, 136, 150, 162catcocl 16613 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧)))
179 f1ocnvfv 6726 . . . . . . . . . 10 ((((𝐹𝑢)𝐺(𝐹𝑧)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))–1-1-onto→(𝑢(Hom ‘𝑌)𝑧) ∧ ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑧))) → ((((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓) → (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))))
180176, 178, 179syl2anc 579 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((((𝐹𝑢)𝐺(𝐹𝑧))‘((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))) = (𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓) → (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))))
181171, 180mpd 15 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)))
182 simpl 474 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑧) → 𝑥 = 𝑢)
183182fveq2d 6379 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑧) → (𝐹𝑥) = (𝐹𝑢))
184 simpr 477 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑧) → 𝑦 = 𝑧)
185184fveq2d 6379 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
186183, 185oveq12d 6860 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑧)))
187186cnveqd 5466 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑧)))
188 ovex 6874 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑧)) ∈ V
189188cnvex 7311 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑧)) ∈ V
190187, 17, 189ovmpt2a 6989 . . . . . . . . . 10 ((𝑢𝑆𝑧𝑆) → (𝑢𝐻𝑧) = ((𝐹𝑢)𝐺(𝐹𝑧)))
191131, 135, 190syl2anc 579 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑢𝐻𝑧) = ((𝐹𝑢)𝐺(𝐹𝑧)))
192191fveq1d 6377 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑧)‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = (((𝐹𝑢)𝐺(𝐹𝑧))‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)))
193 simpl 474 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑣𝑦 = 𝑧) → 𝑥 = 𝑣)
194193fveq2d 6379 . . . . . . . . . . . . . 14 ((𝑥 = 𝑣𝑦 = 𝑧) → (𝐹𝑥) = (𝐹𝑣))
195 simpr 477 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑣𝑦 = 𝑧) → 𝑦 = 𝑧)
196195fveq2d 6379 . . . . . . . . . . . . . 14 ((𝑥 = 𝑣𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
197194, 196oveq12d 6860 . . . . . . . . . . . . 13 ((𝑥 = 𝑣𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑣)𝐺(𝐹𝑧)))
198197cnveqd 5466 . . . . . . . . . . . 12 ((𝑥 = 𝑣𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑣)𝐺(𝐹𝑧)))
199 ovex 6874 . . . . . . . . . . . . 13 ((𝐹𝑣)𝐺(𝐹𝑧)) ∈ V
200199cnvex 7311 . . . . . . . . . . . 12 ((𝐹𝑣)𝐺(𝐹𝑧)) ∈ V
201198, 17, 200ovmpt2a 6989 . . . . . . . . . . 11 ((𝑣𝑆𝑧𝑆) → (𝑣𝐻𝑧) = ((𝐹𝑣)𝐺(𝐹𝑧)))
202133, 135, 201syl2anc 579 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑣𝐻𝑧) = ((𝐹𝑣)𝐺(𝐹𝑧)))
203202fveq1d 6377 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑣𝐻𝑧)‘𝑔) = (((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))
204131, 133, 91syl2anc 579 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
205204fveq1d 6377 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑣)‘𝑓) = (((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))
206203, 205oveq12d 6860 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝑣𝐻𝑧)‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))((𝑢𝐻𝑣)‘𝑓)) = ((((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)))
207181, 192, 2063eqtr4d 2809 . . . . . . 7 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑧)‘(𝑔(⟨𝑢, 𝑣⟩(comp‘𝑌)𝑧)𝑓)) = (((𝑣𝐻𝑧)‘𝑔)(⟨(𝐹𝑢), (𝐹𝑣)⟩(comp‘𝑋)(𝐹𝑧))((𝑢𝐻𝑣)‘𝑓)))
20852, 30, 32, 31, 53, 54, 55, 56, 64, 66, 69, 73, 103, 128, 207isfuncd 16792 . . . . . 6 (𝜑𝐹(𝑌 Func 𝑋)𝐻)
20930, 51, 208cofuval2 16814 . . . . 5 (𝜑 → (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐹𝐹), (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)))⟩)
210 eqid 2765 . . . . . 6 (idfunc𝑋) = (idfunc𝑋)
211210, 30, 66, 31idfuval 16803 . . . . 5 (𝜑 → (idfunc𝑋) = ⟨( I ↾ 𝑅), (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧)))⟩)
21246, 209, 2113eqtr4d 2809 . . . 4 (𝜑 → (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩) = (idfunc𝑋))
213 eqid 2765 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
214 df-br 4810 . . . . . 6 (𝐹(𝑋 Func 𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋 Func 𝑌))
21551, 214sylib 209 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋 Func 𝑌))
216 df-br 4810 . . . . . 6 (𝐹(𝑌 Func 𝑋)𝐻 ↔ ⟨𝐹, 𝐻⟩ ∈ (𝑌 Func 𝑋))
217208, 216sylib 209 . . . . 5 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝑌 Func 𝑋))
21857, 58, 59, 213, 65, 63, 65, 215, 217catcco 17018 . . . 4 (𝜑 → (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩))
219 eqid 2765 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
22057, 58, 219, 210, 59, 65catcid 17020 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
221212, 218, 2203eqtr4d 2809 . . 3 (𝜑 → (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = ((Id‘𝐶)‘𝑋))
222 eqid 2765 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
223 eqid 2765 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
22457catccat 17021 . . . . 5 (𝑈𝑉𝐶 ∈ Cat)
22559, 224syl 17 . . . 4 (𝜑𝐶 ∈ Cat)
22657, 58, 59, 222, 65, 63catchom 17016 . . . . 5 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
227215, 226eleqtrrd 2847 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(Hom ‘𝐶)𝑌))
22857, 58, 59, 222, 63, 65catchom 17016 . . . . 5 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
229217, 228eleqtrrd 2847 . . . 4 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝑌(Hom ‘𝐶)𝑋))
23058, 222, 213, 219, 223, 225, 65, 63, 227, 229issect2 16681 . . 3 (𝜑 → (⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩ ↔ (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = ((Id‘𝐶)‘𝑋)))
231221, 230mpbird 248 . 2 (𝜑 → ⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩)
232 f1ococnv2 6346 . . . . . . 7 (𝐹:𝑅1-1-onto𝑆 → (𝐹𝐹) = ( I ↾ 𝑆))
2331, 232syl 17 . . . . . 6 (𝜑 → (𝐹𝐹) = ( I ↾ 𝑆))
234913adant1 1160 . . . . . . . . . 10 ((𝜑𝑢𝑆𝑣𝑆) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
235234coeq2d 5453 . . . . . . . . 9 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)) = (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))))
236333ad2ant1 1163 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
237753adant3 1162 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → (𝐹𝑢) ∈ 𝑅)
238773adant2 1161 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → (𝐹𝑣) ∈ 𝑅)
23930, 31, 32, 236, 237, 238ffthf1o 16846 . . . . . . . . . . 11 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
2401003impb 1143 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
241240, 143syl 17 . . . . . . . . . . 11 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) ↔ ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣)))
242239, 241mpbid 223 . . . . . . . . . 10 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣))
243 f1ococnv2 6346 . . . . . . . . . 10 (((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
244242, 243syl 17 . . . . . . . . 9 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
245235, 244eqtrd 2799 . . . . . . . 8 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
246245mpt2eq3dva 6917 . . . . . . 7 (𝜑 → (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣))) = (𝑢𝑆, 𝑣𝑆 ↦ ( I ↾ (𝑢(Hom ‘𝑌)𝑣))))
247 fveq2 6375 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑌)‘𝑧) = ((Hom ‘𝑌)‘⟨𝑢, 𝑣⟩))
248 df-ov 6845 . . . . . . . . . 10 (𝑢(Hom ‘𝑌)𝑣) = ((Hom ‘𝑌)‘⟨𝑢, 𝑣⟩)
249247, 248syl6eqr 2817 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑌)‘𝑧) = (𝑢(Hom ‘𝑌)𝑣))
250249reseq2d 5565 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → ( I ↾ ((Hom ‘𝑌)‘𝑧)) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
251250mpt2mpt 6950 . . . . . . 7 (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧))) = (𝑢𝑆, 𝑣𝑆 ↦ ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
252246, 251syl6eqr 2817 . . . . . 6 (𝜑 → (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧))))
253233, 252opeq12d 4567 . . . . 5 (𝜑 → ⟨(𝐹𝐹), (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)))⟩ = ⟨( I ↾ 𝑆), (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧)))⟩)
25452, 208, 51cofuval2 16814 . . . . 5 (𝜑 → (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩) = ⟨(𝐹𝐹), (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)))⟩)
255 eqid 2765 . . . . . 6 (idfunc𝑌) = (idfunc𝑌)
256255, 52, 64, 32idfuval 16803 . . . . 5 (𝜑 → (idfunc𝑌) = ⟨( I ↾ 𝑆), (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧)))⟩)
257253, 254, 2563eqtr4d 2809 . . . 4 (𝜑 → (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩) = (idfunc𝑌))
25857, 58, 59, 213, 63, 65, 63, 217, 215catcco 17018 . . . 4 (𝜑 → (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩))
25957, 58, 219, 255, 59, 63catcid 17020 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
260257, 258, 2593eqtr4d 2809 . . 3 (𝜑 → (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = ((Id‘𝐶)‘𝑌))
26158, 222, 213, 219, 223, 225, 63, 65, 229, 227issect2 16681 . . 3 (𝜑 → (⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩ ↔ (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = ((Id‘𝐶)‘𝑌)))
262260, 261mpbird 248 . 2 (𝜑 → ⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩)
263 catcisolem.i . . 3 𝐼 = (Inv‘𝐶)
26458, 263, 225, 65, 63, 223isinv 16687 . 2 (𝜑 → (⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩ ↔ (⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩ ∧ ⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩)))
265231, 262, 264mpbir2and 704 1 (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cin 3731  cop 4340   class class class wbr 4809  cmpt 4888   I cid 5184   × cxp 5275  ccnv 5276  cres 5279  ccom 5281   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  Basecbs 16132  Hom chom 16227  compcco 16228  Catccat 16592  Idccid 16593  Sectcsect 16671  Invcinv 16672   Func cfunc 16781  idfunccidfu 16782  func ccofu 16783   Full cful 16829   Faith cfth 16830  CatCatccatc 17011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-hom 16240  df-cco 16241  df-cat 16596  df-cid 16597  df-sect 16674  df-inv 16675  df-func 16785  df-idfu 16786  df-cofu 16787  df-full 16831  df-fth 16832  df-catc 17012
This theorem is referenced by:  catciso  17024
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