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Theorem prf2nd 18260
Description: Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prf1st.p 𝑃 = (𝐹 ⟨,⟩F 𝐺)
prf1st.c (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prf1st.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
Assertion
Ref Expression
prf2nd (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺)

Proof of Theorem prf2nd
Dummy variables 𝑓 𝑥 𝑦 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . . . . 7 (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2 eqid 2734 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2734 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
41, 2, 3xpcbas 18233 . . . . . . 7 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5 eqid 2734 . . . . . . 7 (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6 prf1st.c . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 17913 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simprd 495 . . . . . . . 8 (𝜑𝐷 ∈ Cat)
109adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11 prf1st.d . . . . . . . . . 10 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
12 funcrcl 17913 . . . . . . . . . 10 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simprd 495 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
1514adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16 eqid 2734 . . . . . . 7 (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
17 eqid 2734 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
18 relfunc 17912 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
19 1st2ndbr 8065 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 6, 19sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2117, 2, 20funcf1 17916 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
23 relfunc 17912 . . . . . . . . . . 11 Rel (𝐶 Func 𝐸)
24 1st2ndbr 8065 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2523, 11, 24sylancr 587 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2617, 3, 25funcf1 17916 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
2726ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
2822, 27opelxpd 5727 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
291, 4, 5, 10, 15, 16, 282ndf1 18250 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = (2nd ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
30 fvex 6919 . . . . . . 7 ((1st𝐹)‘𝑥) ∈ V
31 fvex 6919 . . . . . . 7 ((1st𝐺)‘𝑥) ∈ V
3230, 31op2nd 8021 . . . . . 6 (2nd ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐺)‘𝑥)
3329, 32eqtrdi 2790 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐺)‘𝑥))
3433mpteq2dva 5247 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
35 prf1st.p . . . . . . 7 𝑃 = (𝐹 ⟨,⟩F 𝐺)
36 eqid 2734 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3735, 17, 36, 6, 11prfval 18254 . . . . . 6 (𝜑𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
38 fvex 6919 . . . . . . . 8 (Base‘𝐶) ∈ V
3938mptex 7242 . . . . . . 7 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
4038, 38mpoex 8102 . . . . . . 7 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
4139, 40op1std 8022 . . . . . 6 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
4237, 41syl 17 . . . . 5 (𝜑 → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
43 relfunc 17912 . . . . . . . 8 Rel ((𝐷 ×c 𝐸) Func 𝐸)
441, 9, 14, 162ndfcl 18253 . . . . . . . 8 (𝜑 → (𝐷 2ndF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐸))
45 1st2ndbr 8065 . . . . . . . 8 ((Rel ((𝐷 ×c 𝐸) Func 𝐸) ∧ (𝐷 2ndF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐸)) → (1st ‘(𝐷 2ndF 𝐸))((𝐷 ×c 𝐸) Func 𝐸)(2nd ‘(𝐷 2ndF 𝐸)))
4643, 44, 45sylancr 587 . . . . . . 7 (𝜑 → (1st ‘(𝐷 2ndF 𝐸))((𝐷 ×c 𝐸) Func 𝐸)(2nd ‘(𝐷 2ndF 𝐸)))
474, 3, 46funcf1 17916 . . . . . 6 (𝜑 → (1st ‘(𝐷 2ndF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐸))
4847feqmptd 6976 . . . . 5 (𝜑 → (1st ‘(𝐷 2ndF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 2ndF 𝐸))‘𝑢)))
49 fveq2 6906 . . . . 5 (𝑢 = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ → ((1st ‘(𝐷 2ndF 𝐸))‘𝑢) = ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
5028, 42, 48, 49fmptco 7148 . . . 4 (𝜑 → ((1st ‘(𝐷 2ndF 𝐸)) ∘ (1st𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)))
5126feqmptd 6976 . . . 4 (𝜑 → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
5234, 50, 513eqtr4d 2784 . . 3 (𝜑 → ((1st ‘(𝐷 2ndF 𝐸)) ∘ (1st𝑃)) = (1st𝐺))
539ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
5414ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
55 relfunc 17912 . . . . . . . . . . . . . . . 16 Rel (𝐶 Func (𝐷 ×c 𝐸))
5635, 1, 6, 11prfcl 18258 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
57 1st2ndbr 8065 . . . . . . . . . . . . . . . 16 ((Rel (𝐶 Func (𝐷 ×c 𝐸)) ∧ 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5855, 56, 57sylancr 587 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5917, 4, 58funcf1 17916 . . . . . . . . . . . . . 14 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
6059ffvelcdmda 7103 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6261adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6359ffvelcdmda 7103 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6463adantrl 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6564adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
661, 4, 5, 53, 54, 16, 62, 652ndf2 18251 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) = (2nd ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))))
6766fveq1d 6908 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((2nd ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))‘((𝑥(2nd𝑃)𝑦)‘𝑓)))
6858adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
69 simprl 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
70 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
7117, 36, 5, 68, 69, 70funcf2 17918 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))
7271ffvelcdmda 7103 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝑃)𝑦)‘𝑓) ∈ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))
7372fvresd 6926 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((2nd ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = (2nd ‘((𝑥(2nd𝑃)𝑦)‘𝑓)))
746ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
7511ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
7669adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
7770adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
78 simpr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
7935, 17, 36, 74, 75, 76, 77, 78prf2 18257 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝑃)𝑦)‘𝑓) = ⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩)
8079fveq2d 6910 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (2nd ‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = (2nd ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩))
81 fvex 6919 . . . . . . . . . . 11 ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ V
82 fvex 6919 . . . . . . . . . . 11 ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ V
8381, 82op2nd 8021 . . . . . . . . . 10 (2nd ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd𝐺)𝑦)‘𝑓)
8480, 83eqtrdi 2790 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (2nd ‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((𝑥(2nd𝐺)𝑦)‘𝑓))
8567, 73, 843eqtrd 2778 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((𝑥(2nd𝐺)𝑦)‘𝑓))
8685mpteq2dva 5247 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓))) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑓)))
87 eqid 2734 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
8846adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐷 2ndF 𝐸))((𝐷 ×c 𝐸) Func 𝐸)(2nd ‘(𝐷 2ndF 𝐸)))
894, 5, 87, 88, 61, 64funcf2 17918 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)):(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))⟶(((1st ‘(𝐷 2ndF 𝐸))‘((1st𝑃)‘𝑥))(Hom ‘𝐸)((1st ‘(𝐷 2ndF 𝐸))‘((1st𝑃)‘𝑦))))
90 fcompt 7152 . . . . . . . 8 (((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)):(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))⟶(((1st ‘(𝐷 2ndF 𝐸))‘((1st𝑃)‘𝑥))(Hom ‘𝐸)((1st ‘(𝐷 2ndF 𝐸))‘((1st𝑃)‘𝑦))) ∧ (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓))))
9189, 71, 90syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓))))
9225adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
9317, 36, 87, 92, 69, 70funcf2 17918 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
9493feqmptd 6976 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑓)))
9586, 91, 943eqtr4d 2784 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐺)𝑦))
96953impb 1114 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐺)𝑦))
9796mpoeq3dva 7509 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
9817, 25funcfn2 17919 . . . . 5 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
99 fnov 7563 . . . . 5 ((2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
10098, 99sylib 218 . . . 4 (𝜑 → (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
10197, 100eqtr4d 2777 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (2nd𝐺))
10252, 101opeq12d 4885 . 2 (𝜑 → ⟨((1st ‘(𝐷 2ndF 𝐸)) ∘ (1st𝑃)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)))⟩ = ⟨(1st𝐺), (2nd𝐺)⟩)
10317, 56, 44cofuval 17932 . 2 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = ⟨((1st ‘(𝐷 2ndF 𝐸)) ∘ (1st𝑃)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)))⟩)
104 1st2nd 8062 . . 3 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
10523, 11, 104sylancr 587 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
106102, 103, 1053eqtr4d 2784 1 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cop 4636   class class class wbr 5147  cmpt 5230   × cxp 5686  cres 5690  ccom 5692  Rel wrel 5693   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  1st c1st 8010  2nd c2nd 8011  Basecbs 17244  Hom chom 17308  Catccat 17708   Func cfunc 17904  func ccofu 17906   ×c cxpc 18223   2ndF c2ndf 18225   ⟨,⟩F cprf 18226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-cat 17712  df-cid 17713  df-func 17908  df-cofu 17910  df-xpc 18227  df-2ndf 18229  df-prf 18230
This theorem is referenced by: (None)
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