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Theorem catcisolem 16803
Description: Lemma for catciso 16804. (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 6203 . . . . . . 7 (𝐹:𝑅1-1-onto𝑆 → (𝐹𝐹) = ( I ↾ 𝑅))
31, 2syl 17 . . . . . 6 (𝜑 → (𝐹𝐹) = ( I ↾ 𝑅))
413ad2ant1 1102 . . . . . . . . . . . . . 14 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹:𝑅1-1-onto𝑆)
5 f1of 6175 . . . . . . . . . . . . . 14 (𝐹:𝑅1-1-onto𝑆𝐹:𝑅𝑆)
64, 5syl 17 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹:𝑅𝑆)
7 simp2 1082 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝑢𝑅)
86, 7ffvelrnd 6400 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹𝑢) ∈ 𝑆)
9 simp3 1083 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → 𝑣𝑅)
106, 9ffvelrnd 6400 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹𝑣) ∈ 𝑆)
11 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → 𝑥 = (𝐹𝑢))
1211fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → (𝐹𝑥) = (𝐹‘(𝐹𝑢)))
13 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → 𝑦 = (𝐹𝑣))
1413fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → (𝐹𝑦) = (𝐹‘(𝐹𝑣)))
1512, 14oveq12d 6708 . . . . . . . . . . . . . 14 ((𝑥 = (𝐹𝑢) ∧ 𝑦 = (𝐹𝑣)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
1615cnveqd 5330 . . . . . . . . . . . . 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 6718 . . . . . . . . . . . . . 14 ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) ∈ V
1918cnvex 7155 . . . . . . . . . . . . 13 ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) ∈ V
2016, 17, 19ovmpt2a 6833 . . . . . . . . . . . 12 (((𝐹𝑢) ∈ 𝑆 ∧ (𝐹𝑣) ∈ 𝑆) → ((𝐹𝑢)𝐻(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
218, 10, 20syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹𝑢)𝐻(𝐹𝑣)) = ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))))
22 f1ocnvfv1 6572 . . . . . . . . . . . . . 14 ((𝐹:𝑅1-1-onto𝑆𝑢𝑅) → (𝐹‘(𝐹𝑢)) = 𝑢)
234, 7, 22syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹‘(𝐹𝑢)) = 𝑢)
24 f1ocnvfv1 6572 . . . . . . . . . . . . . 14 ((𝐹:𝑅1-1-onto𝑆𝑣𝑅) → (𝐹‘(𝐹𝑣)) = 𝑣)
254, 9, 24syl2anc 694 . . . . . . . . . . . . 13 ((𝜑𝑢𝑅𝑣𝑅) → (𝐹‘(𝐹𝑣)) = 𝑣)
2623, 25oveq12d 6708 . . . . . . . . . . . 12 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) = (𝑢𝐺𝑣))
2726cnveqd 5330 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹‘(𝐹𝑢))𝐺(𝐹‘(𝐹𝑣))) = (𝑢𝐺𝑣))
2821, 27eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑢𝑅𝑣𝑅) → ((𝐹𝑢)𝐻(𝐹𝑣)) = (𝑢𝐺𝑣))
2928coeq1d 5316 . . . . . . . . 9 ((𝜑𝑢𝑅𝑣𝑅) → (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)) = ((𝑢𝐺𝑣) ∘ (𝑢𝐺𝑣)))
30 catciso.r . . . . . . . . . . 11 𝑅 = (Base‘𝑋)
31 eqid 2651 . . . . . . . . . . 11 (Hom ‘𝑋) = (Hom ‘𝑋)
32 eqid 2651 . . . . . . . . . . 11 (Hom ‘𝑌) = (Hom ‘𝑌)
33 catcisolem.1 . . . . . . . . . . . 12 (𝜑𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
34333ad2ant1 1102 . . . . . . . . . . 11 ((𝜑𝑢𝑅𝑣𝑅) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
3530, 31, 32, 34, 7, 9ffthf1o 16626 . . . . . . . . . 10 ((𝜑𝑢𝑅𝑣𝑅) → (𝑢𝐺𝑣):(𝑢(Hom ‘𝑋)𝑣)–1-1-onto→((𝐹𝑢)(Hom ‘𝑌)(𝐹𝑣)))
36 f1ococnv1 6203 . . . . . . . . . 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 2685 . . . . . . . 8 ((𝜑𝑢𝑅𝑣𝑅) → (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
3938mpt2eq3dva 6761 . . . . . . 7 (𝜑 → (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣))) = (𝑢𝑅, 𝑣𝑅 ↦ ( I ↾ (𝑢(Hom ‘𝑋)𝑣))))
40 fveq2 6229 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑋)‘𝑧) = ((Hom ‘𝑋)‘⟨𝑢, 𝑣⟩))
41 df-ov 6693 . . . . . . . . . 10 (𝑢(Hom ‘𝑋)𝑣) = ((Hom ‘𝑋)‘⟨𝑢, 𝑣⟩)
4240, 41syl6eqr 2703 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑋)‘𝑧) = (𝑢(Hom ‘𝑋)𝑣))
4342reseq2d 5428 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → ( I ↾ ((Hom ‘𝑋)‘𝑧)) = ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
4443mpt2mpt 6794 . . . . . . 7 (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧))) = (𝑢𝑅, 𝑣𝑅 ↦ ( I ↾ (𝑢(Hom ‘𝑋)𝑣)))
4539, 44syl6eqr 2703 . . . . . 6 (𝜑 → (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣))) = (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧))))
463, 45opeq12d 4441 . . . . 5 (𝜑 → ⟨(𝐹𝐹), (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)))⟩ = ⟨( I ↾ 𝑅), (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧)))⟩)
47 inss1 3866 . . . . . . . . 9 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Full 𝑌)
48 fullfunc 16613 . . . . . . . . 9 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
4947, 48sstri 3645 . . . . . . . 8 ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ⊆ (𝑋 Func 𝑌)
5049ssbri 4730 . . . . . . 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 2651 . . . . . . 7 (Id‘𝑌) = (Id‘𝑌)
54 eqid 2651 . . . . . . 7 (Id‘𝑋) = (Id‘𝑋)
55 eqid 2651 . . . . . . 7 (comp‘𝑌) = (comp‘𝑌)
56 eqid 2651 . . . . . . 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 16794 . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Cat))
61 inss2 3867 . . . . . . . . 9 (𝑈 ∩ Cat) ⊆ Cat
6260, 61syl6eqss 3688 . . . . . . . 8 (𝜑𝐵 ⊆ Cat)
63 catciso.y . . . . . . . 8 (𝜑𝑌𝐵)
6462, 63sseldd 3637 . . . . . . 7 (𝜑𝑌 ∈ Cat)
65 catciso.x . . . . . . . 8 (𝜑𝑋𝐵)
6662, 65sseldd 3637 . . . . . . 7 (𝜑𝑋 ∈ Cat)
67 f1ocnv 6187 . . . . . . . 8 (𝐹:𝑅1-1-onto𝑆𝐹:𝑆1-1-onto𝑅)
68 f1of 6175 . . . . . . . 8 (𝐹:𝑆1-1-onto𝑅𝐹:𝑆𝑅)
691, 67, 683syl 18 . . . . . . 7 (𝜑𝐹:𝑆𝑅)
70 ovex 6718 . . . . . . . . . 10 ((𝐹𝑥)𝐺(𝐹𝑦)) ∈ V
7170cnvex 7155 . . . . . . . . 9 ((𝐹𝑥)𝐺(𝐹𝑦)) ∈ V
7217, 71fnmpt2i 7284 . . . . . . . 8 𝐻 Fn (𝑆 × 𝑆)
7372a1i 11 . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
7433adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
7569ffvelrnda 6399 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → (𝐹𝑢) ∈ 𝑅)
7675adantrr 753 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹𝑢) ∈ 𝑅)
7769ffvelrnda 6399 . . . . . . . . . . 11 ((𝜑𝑣𝑆) → (𝐹𝑣) ∈ 𝑅)
7877adantrl 752 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹𝑣) ∈ 𝑅)
7930, 31, 32, 74, 76, 78ffthf1o 16626 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
80 f1ocnv 6187 . . . . . . . . 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 6175 . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
8483fveq2d 6233 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝐹𝑥) = (𝐹𝑢))
85 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
8685fveq2d 6233 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑣) → (𝐹𝑦) = (𝐹𝑣))
8784, 86oveq12d 6708 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑣)))
8887cnveqd 5330 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑣) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑣)))
89 ovex 6718 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑣)) ∈ V
9089cnvex 7155 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑣)) ∈ V
9188, 17, 90ovmpt2a 6833 . . . . . . . . . 10 ((𝑢𝑆𝑣𝑆) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
9291adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
931adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝐹:𝑅1-1-onto𝑆)
94 simprl 809 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝑢𝑆)
95 f1ocnvfv2 6573 . . . . . . . . . . . 12 ((𝐹:𝑅1-1-onto𝑆𝑢𝑆) → (𝐹‘(𝐹𝑢)) = 𝑢)
9693, 94, 95syl2anc 694 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹‘(𝐹𝑢)) = 𝑢)
97 simprr 811 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → 𝑣𝑆)
98 f1ocnvfv2 6573 . . . . . . . . . . . 12 ((𝐹:𝑅1-1-onto𝑆𝑣𝑆) → (𝐹‘(𝐹𝑣)) = 𝑣)
9993, 97, 98syl2anc 694 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝐹‘(𝐹𝑣)) = 𝑣)
10096, 99oveq12d 6708 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))) = (𝑢(Hom ‘𝑌)𝑣))
101100eqcomd 2657 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢(Hom ‘𝑌)𝑣) = ((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
10292, 101feq12d 6071 . . . . . . . 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 247 . . . . . . 7 ((𝜑 ∧ (𝑢𝑆𝑣𝑆)) → (𝑢𝐻𝑣):(𝑢(Hom ‘𝑌)𝑣)⟶((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣)))
104 simpr 476 . . . . . . . . . 10 ((𝜑𝑢𝑆) → 𝑢𝑆)
105 simpl 472 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑥 = 𝑢)
106105fveq2d 6233 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝐹𝑥) = (𝐹𝑢))
107 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑢) → 𝑦 = 𝑢)
108107fveq2d 6233 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑢) → (𝐹𝑦) = (𝐹𝑢))
109106, 108oveq12d 6708 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑢)))
110109cnveqd 5330 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑢) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑢)))
111 ovex 6718 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑢)) ∈ V
112111cnvex 7155 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑢)) ∈ V
113110, 17, 112ovmpt2a 6833 . . . . . . . . . 10 ((𝑢𝑆𝑢𝑆) → (𝑢𝐻𝑢) = ((𝐹𝑢)𝐺(𝐹𝑢)))
114104, 104, 113syl2anc 694 . . . . . . . . 9 ((𝜑𝑢𝑆) → (𝑢𝐻𝑢) = ((𝐹𝑢)𝐺(𝐹𝑢)))
115114fveq1d 6231 . . . . . . . 8 ((𝜑𝑢𝑆) → ((𝑢𝐻𝑢)‘((Id‘𝑌)‘𝑢)) = (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑌)‘𝑢)))
11651adantr 480 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝐹(𝑋 Func 𝑌)𝐺)
11730, 54, 53, 116, 75funcid 16577 . . . . . . . . . 10 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘(𝐹‘(𝐹𝑢))))
1181, 95sylan 487 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → (𝐹‘(𝐹𝑢)) = 𝑢)
119118fveq2d 6233 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((Id‘𝑌)‘(𝐹‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢))
120117, 119eqtrd 2685 . . . . . . . . 9 ((𝜑𝑢𝑆) → (((𝐹𝑢)𝐺(𝐹𝑢))‘((Id‘𝑋)‘(𝐹𝑢))) = ((Id‘𝑌)‘𝑢))
12133adantr 480 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
12230, 31, 32, 121, 75, 75ffthf1o 16626 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((𝐹𝑢)𝐺(𝐹𝑢)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑢))))
12366adantr 480 . . . . . . . . . . 11 ((𝜑𝑢𝑆) → 𝑋 ∈ Cat)
12430, 31, 54, 123, 75catidcl 16390 . . . . . . . . . 10 ((𝜑𝑢𝑆) → ((Id‘𝑋)‘(𝐹𝑢)) ∈ ((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑢)))
125 f1ocnvfv 6574 . . . . . . . . . 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 694 . . . . . . . . 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 2685 . . . . . . 7 ((𝜑𝑢𝑆) → ((𝑢𝐻𝑢)‘((Id‘𝑌)‘𝑢)) = ((Id‘𝑋)‘(𝐹𝑢)))
129513ad2ant1 1102 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹(𝑋 Func 𝑌)𝐺)
130693ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹:𝑆𝑅)
131 simp21 1114 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑢𝑆)
132130, 131ffvelrnd 6400 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑢) ∈ 𝑅)
133 simp22 1115 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑣𝑆)
134130, 133ffvelrnd 6400 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑣) ∈ 𝑅)
135 simp23 1116 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑧𝑆)
136130, 135ffvelrnd 6400 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹𝑧) ∈ 𝑅)
137333ad2ant1 1102 . . . . . . . . . . . . . . 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 16626 . . . . . . . . . . . . . 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 1102 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝐹:𝑅1-1-onto𝑆)
140139, 131, 95syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑢)) = 𝑢)
141139, 133, 98syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑣)) = 𝑣)
142140, 141oveq12d 6708 . . . . . . . . . . . . . . 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 6167 . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 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 6187 . . . . . . . . . . . . 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 6175 . . . . . . . . . . . . 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 1109 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣))
150148, 149ffvelrnd 6400 . . . . . . . . . . 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 16626 . . . . . . . . . . . . . 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 6573 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑅1-1-onto𝑆𝑧𝑆) → (𝐹‘(𝐹𝑧)) = 𝑧)
153139, 135, 152syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
154141, 153oveq12d 6708 . . . . . . . . . . . . . . 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 6167 . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 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 6187 . . . . . . . . . . . . 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 6175 . . . . . . . . . . . . 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 1110 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))
162160, 161ffvelrnd 6400 . . . . . . . . . . 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 16578 . . . . . . . . . 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 4441 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ⟨(𝐹‘(𝐹𝑢)), (𝐹‘(𝐹𝑣))⟩ = ⟨𝑢, 𝑣⟩)
165164, 153oveq12d 6708 . . . . . . . . . . 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 6573 . . . . . . . . . . . 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 694 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑣)𝐺(𝐹𝑧))‘(((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔)) = 𝑔)
168 f1ocnvfv2 6573 . . . . . . . . . . . 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 694 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (((𝐹𝑢)𝐺(𝐹𝑣))‘(((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓)) = 𝑓)
170165, 167, 169oveq123d 6711 . . . . . . . . . 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 2685 . . . . . . . . 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 16626 . . . . . . . . . . 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 6708 . . . . . . . . . . . 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 6167 . . . . . . . . . . . 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 222 . . . . . . . . . 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 1102 . . . . . . . . . . 11 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → 𝑋 ∈ Cat)
17830, 31, 56, 177, 132, 134, 136, 150, 162catcocl 16393 . . . . . . . . . 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 6574 . . . . . . . . . 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 694 . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑧) → 𝑥 = 𝑢)
183182fveq2d 6233 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑧) → (𝐹𝑥) = (𝐹𝑢))
184 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = 𝑢𝑦 = 𝑧) → 𝑦 = 𝑧)
185184fveq2d 6233 . . . . . . . . . . . . 13 ((𝑥 = 𝑢𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
186183, 185oveq12d 6708 . . . . . . . . . . . 12 ((𝑥 = 𝑢𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑧)))
187186cnveqd 5330 . . . . . . . . . . 11 ((𝑥 = 𝑢𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑢)𝐺(𝐹𝑧)))
188 ovex 6718 . . . . . . . . . . . 12 ((𝐹𝑢)𝐺(𝐹𝑧)) ∈ V
189188cnvex 7155 . . . . . . . . . . 11 ((𝐹𝑢)𝐺(𝐹𝑧)) ∈ V
190187, 17, 189ovmpt2a 6833 . . . . . . . . . 10 ((𝑢𝑆𝑧𝑆) → (𝑢𝐻𝑧) = ((𝐹𝑢)𝐺(𝐹𝑧)))
191131, 135, 190syl2anc 694 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑢𝐻𝑧) = ((𝐹𝑢)𝐺(𝐹𝑧)))
192191fveq1d 6231 . . . . . . . 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 472 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑣𝑦 = 𝑧) → 𝑥 = 𝑣)
194193fveq2d 6233 . . . . . . . . . . . . . 14 ((𝑥 = 𝑣𝑦 = 𝑧) → (𝐹𝑥) = (𝐹𝑣))
195 simpr 476 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑣𝑦 = 𝑧) → 𝑦 = 𝑧)
196195fveq2d 6233 . . . . . . . . . . . . . 14 ((𝑥 = 𝑣𝑦 = 𝑧) → (𝐹𝑦) = (𝐹𝑧))
197194, 196oveq12d 6708 . . . . . . . . . . . . 13 ((𝑥 = 𝑣𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑣)𝐺(𝐹𝑧)))
198197cnveqd 5330 . . . . . . . . . . . 12 ((𝑥 = 𝑣𝑦 = 𝑧) → ((𝐹𝑥)𝐺(𝐹𝑦)) = ((𝐹𝑣)𝐺(𝐹𝑧)))
199 ovex 6718 . . . . . . . . . . . . 13 ((𝐹𝑣)𝐺(𝐹𝑧)) ∈ V
200199cnvex 7155 . . . . . . . . . . . 12 ((𝐹𝑣)𝐺(𝐹𝑧)) ∈ V
201198, 17, 200ovmpt2a 6833 . . . . . . . . . . 11 ((𝑣𝑆𝑧𝑆) → (𝑣𝐻𝑧) = ((𝐹𝑣)𝐺(𝐹𝑧)))
202133, 135, 201syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑣𝐻𝑧) = ((𝐹𝑣)𝐺(𝐹𝑧)))
203202fveq1d 6231 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑣𝐻𝑧)‘𝑔) = (((𝐹𝑣)𝐺(𝐹𝑧))‘𝑔))
204131, 133, 91syl2anc 694 . . . . . . . . . 10 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
205204fveq1d 6231 . . . . . . . . 9 ((𝜑 ∧ (𝑢𝑆𝑣𝑆𝑧𝑆) ∧ (𝑓 ∈ (𝑢(Hom ‘𝑌)𝑣) ∧ 𝑔 ∈ (𝑣(Hom ‘𝑌)𝑧))) → ((𝑢𝐻𝑣)‘𝑓) = (((𝐹𝑢)𝐺(𝐹𝑣))‘𝑓))
206203, 205oveq12d 6708 . . . . . . . 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 2695 . . . . . . 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 16572 . . . . . 6 (𝜑𝐹(𝑌 Func 𝑋)𝐻)
20930, 51, 208cofuval2 16594 . . . . 5 (𝜑 → (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩) = ⟨(𝐹𝐹), (𝑢𝑅, 𝑣𝑅 ↦ (((𝐹𝑢)𝐻(𝐹𝑣)) ∘ (𝑢𝐺𝑣)))⟩)
210 eqid 2651 . . . . . 6 (idfunc𝑋) = (idfunc𝑋)
211210, 30, 66, 31idfuval 16583 . . . . 5 (𝜑 → (idfunc𝑋) = ⟨( I ↾ 𝑅), (𝑧 ∈ (𝑅 × 𝑅) ↦ ( I ↾ ((Hom ‘𝑋)‘𝑧)))⟩)
21246, 209, 2113eqtr4d 2695 . . . 4 (𝜑 → (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩) = (idfunc𝑋))
213 eqid 2651 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
214 df-br 4686 . . . . . 6 (𝐹(𝑋 Func 𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋 Func 𝑌))
21551, 214sylib 208 . . . . 5 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋 Func 𝑌))
216 df-br 4686 . . . . . 6 (𝐹(𝑌 Func 𝑋)𝐻 ↔ ⟨𝐹, 𝐻⟩ ∈ (𝑌 Func 𝑋))
217208, 216sylib 208 . . . . 5 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝑌 Func 𝑋))
21857, 58, 59, 213, 65, 63, 65, 215, 217catcco 16798 . . . 4 (𝜑 → (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = (⟨𝐹, 𝐻⟩ ∘func𝐹, 𝐺⟩))
219 eqid 2651 . . . . 5 (Id‘𝐶) = (Id‘𝐶)
22057, 58, 219, 210, 59, 65catcid 16800 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑋) = (idfunc𝑋))
221212, 218, 2203eqtr4d 2695 . . 3 (𝜑 → (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = ((Id‘𝐶)‘𝑋))
222 eqid 2651 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
223 eqid 2651 . . . 4 (Sect‘𝐶) = (Sect‘𝐶)
22457catccat 16801 . . . . 5 (𝑈𝑉𝐶 ∈ Cat)
22559, 224syl 17 . . . 4 (𝜑𝐶 ∈ Cat)
22657, 58, 59, 222, 65, 63catchom 16796 . . . . 5 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 Func 𝑌))
227215, 226eleqtrrd 2733 . . . 4 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(Hom ‘𝐶)𝑌))
22857, 58, 59, 222, 63, 65catchom 16796 . . . . 5 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 Func 𝑋))
229217, 228eleqtrrd 2733 . . . 4 (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝑌(Hom ‘𝐶)𝑋))
23058, 222, 213, 219, 223, 225, 65, 63, 227, 229issect2 16461 . . 3 (𝜑 → (⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩ ↔ (⟨𝐹, 𝐻⟩(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)⟨𝐹, 𝐺⟩) = ((Id‘𝐶)‘𝑋)))
231221, 230mpbird 247 . 2 (𝜑 → ⟨𝐹, 𝐺⟩(𝑋(Sect‘𝐶)𝑌)⟨𝐹, 𝐻⟩)
232 f1ococnv2 6201 . . . . . . 7 (𝐹:𝑅1-1-onto𝑆 → (𝐹𝐹) = ( I ↾ 𝑆))
2331, 232syl 17 . . . . . 6 (𝜑 → (𝐹𝐹) = ( I ↾ 𝑆))
234913adant1 1099 . . . . . . . . . 10 ((𝜑𝑢𝑆𝑣𝑆) → (𝑢𝐻𝑣) = ((𝐹𝑢)𝐺(𝐹𝑣)))
235234coeq2d 5317 . . . . . . . . 9 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)) = (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ ((𝐹𝑢)𝐺(𝐹𝑣))))
236333ad2ant1 1102 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → 𝐹((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌))𝐺)
237753adant3 1101 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → (𝐹𝑢) ∈ 𝑅)
238773adant2 1100 . . . . . . . . . . . 12 ((𝜑𝑢𝑆𝑣𝑆) → (𝐹𝑣) ∈ 𝑅)
23930, 31, 32, 236, 237, 238ffthf1o 16626 . . . . . . . . . . 11 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→((𝐹‘(𝐹𝑢))(Hom ‘𝑌)(𝐹‘(𝐹𝑣))))
2401003impb 1279 . . . . . . . . . . . 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 222 . . . . . . . . . 10 ((𝜑𝑢𝑆𝑣𝑆) → ((𝐹𝑢)𝐺(𝐹𝑣)):((𝐹𝑢)(Hom ‘𝑋)(𝐹𝑣))–1-1-onto→(𝑢(Hom ‘𝑌)𝑣))
243 f1ococnv2 6201 . . . . . . . . . 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 2685 . . . . . . . 8 ((𝜑𝑢𝑆𝑣𝑆) → (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
246245mpt2eq3dva 6761 . . . . . . 7 (𝜑 → (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣))) = (𝑢𝑆, 𝑣𝑆 ↦ ( I ↾ (𝑢(Hom ‘𝑌)𝑣))))
247 fveq2 6229 . . . . . . . . . 10 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑌)‘𝑧) = ((Hom ‘𝑌)‘⟨𝑢, 𝑣⟩))
248 df-ov 6693 . . . . . . . . . 10 (𝑢(Hom ‘𝑌)𝑣) = ((Hom ‘𝑌)‘⟨𝑢, 𝑣⟩)
249247, 248syl6eqr 2703 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ → ((Hom ‘𝑌)‘𝑧) = (𝑢(Hom ‘𝑌)𝑣))
250249reseq2d 5428 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → ( I ↾ ((Hom ‘𝑌)‘𝑧)) = ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
251250mpt2mpt 6794 . . . . . . 7 (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧))) = (𝑢𝑆, 𝑣𝑆 ↦ ( I ↾ (𝑢(Hom ‘𝑌)𝑣)))
252246, 251syl6eqr 2703 . . . . . 6 (𝜑 → (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧))))
253233, 252opeq12d 4441 . . . . 5 (𝜑 → ⟨(𝐹𝐹), (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)))⟩ = ⟨( I ↾ 𝑆), (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧)))⟩)
25452, 208, 51cofuval2 16594 . . . . 5 (𝜑 → (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩) = ⟨(𝐹𝐹), (𝑢𝑆, 𝑣𝑆 ↦ (((𝐹𝑢)𝐺(𝐹𝑣)) ∘ (𝑢𝐻𝑣)))⟩)
255 eqid 2651 . . . . . 6 (idfunc𝑌) = (idfunc𝑌)
256255, 52, 64, 32idfuval 16583 . . . . 5 (𝜑 → (idfunc𝑌) = ⟨( I ↾ 𝑆), (𝑧 ∈ (𝑆 × 𝑆) ↦ ( I ↾ ((Hom ‘𝑌)‘𝑧)))⟩)
257253, 254, 2563eqtr4d 2695 . . . 4 (𝜑 → (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩) = (idfunc𝑌))
25857, 58, 59, 213, 63, 65, 63, 217, 215catcco 16798 . . . 4 (𝜑 → (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = (⟨𝐹, 𝐺⟩ ∘func𝐹, 𝐻⟩))
25957, 58, 219, 255, 59, 63catcid 16800 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) = (idfunc𝑌))
260257, 258, 2593eqtr4d 2695 . . 3 (𝜑 → (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = ((Id‘𝐶)‘𝑌))
26158, 222, 213, 219, 223, 225, 63, 65, 229, 227issect2 16461 . . 3 (𝜑 → (⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩ ↔ (⟨𝐹, 𝐺⟩(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)⟨𝐹, 𝐻⟩) = ((Id‘𝐶)‘𝑌)))
262260, 261mpbird 247 . 2 (𝜑 → ⟨𝐹, 𝐻⟩(𝑌(Sect‘𝐶)𝑋)⟨𝐹, 𝐺⟩)
263 catcisolem.i . . 3 𝐼 = (Inv‘𝐶)
26458, 263, 225, 65, 63, 223isinv 16467 . 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 977 1 (𝜑 → ⟨𝐹, 𝐺⟩(𝑋𝐼𝑌)⟨𝐹, 𝐻⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  cin 3606  cop 4216   class class class wbr 4685  cmpt 4762   I cid 5052   × cxp 5141  ccnv 5142  cres 5145  ccom 5147   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Sectcsect 16451  Invcinv 16452   Func cfunc 16561  idfunccidfu 16562  func ccofu 16563   Full cful 16609   Faith cfth 16610  CatCatccatc 16791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-sect 16454  df-inv 16455  df-func 16565  df-idfu 16566  df-cofu 16567  df-full 16611  df-fth 16612  df-catc 16792
This theorem is referenced by:  catciso  16804
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