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Theorem prf1st 17520
 Description: Cancellation of pairing with first 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
prf1st (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)

Proof of Theorem prf1st
Dummy variables 𝑓 𝑥 𝑦 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2758 . . . . . . 7 (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2 eqid 2758 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2758 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
41, 2, 3xpcbas 17494 . . . . . . 7 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5 eqid 2758 . . . . . . 7 (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6 prf1st.c . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 17192 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simprd 499 . . . . . . . 8 (𝜑𝐷 ∈ Cat)
109adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11 prf1st.d . . . . . . . . . 10 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
12 funcrcl 17192 . . . . . . . . . 10 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simprd 499 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
1514adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16 eqid 2758 . . . . . . 7 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
17 eqid 2758 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
18 relfunc 17191 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
19 1st2ndbr 7745 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 6, 19sylancr 590 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2117, 2, 20funcf1 17195 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelrnda 6842 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
23 relfunc 17191 . . . . . . . . . . 11 Rel (𝐶 Func 𝐸)
24 1st2ndbr 7745 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2523, 11, 24sylancr 590 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2617, 3, 25funcf1 17195 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
2726ffvelrnda 6842 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
2822, 27opelxpd 5562 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
291, 4, 5, 10, 15, 16, 281stf1 17508 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = (1st ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
30 fvex 6671 . . . . . . 7 ((1st𝐹)‘𝑥) ∈ V
31 fvex 6671 . . . . . . 7 ((1st𝐺)‘𝑥) ∈ V
3230, 31op1st 7701 . . . . . 6 (1st ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐹)‘𝑥)
3329, 32eqtrdi 2809 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐹)‘𝑥))
3433mpteq2dva 5127 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
35 prf1st.p . . . . . . 7 𝑃 = (𝐹 ⟨,⟩F 𝐺)
36 eqid 2758 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3735, 17, 36, 6, 11prfval 17515 . . . . . 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 6671 . . . . . . . 8 (Base‘𝐶) ∈ V
3938mptex 6977 . . . . . . 7 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
4038, 38mpoex 7782 . . . . . . 7 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
4139, 40op1std 7703 . . . . . 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 17191 . . . . . . . 8 Rel ((𝐷 ×c 𝐸) Func 𝐷)
441, 9, 14, 161stfcl 17513 . . . . . . . 8 (𝜑 → (𝐷 1stF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐷))
45 1st2ndbr 7745 . . . . . . . 8 ((Rel ((𝐷 ×c 𝐸) Func 𝐷) ∧ (𝐷 1stF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐷)) → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
4643, 44, 45sylancr 590 . . . . . . 7 (𝜑 → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
474, 2, 46funcf1 17195 . . . . . 6 (𝜑 → (1st ‘(𝐷 1stF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐷))
4847feqmptd 6721 . . . . 5 (𝜑 → (1st ‘(𝐷 1stF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 1stF 𝐸))‘𝑢)))
49 fveq2 6658 . . . . 5 (𝑢 = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ → ((1st ‘(𝐷 1stF 𝐸))‘𝑢) = ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
5028, 42, 48, 49fmptco 6882 . . . 4 (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)))
5121feqmptd 6721 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
5234, 50, 513eqtr4d 2803 . . 3 (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)) = (1st𝐹))
539ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
5414ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
55 relfunc 17191 . . . . . . . . . . . . . . . 16 Rel (𝐶 Func (𝐷 ×c 𝐸))
5635, 1, 6, 11prfcl 17519 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
57 1st2ndbr 7745 . . . . . . . . . . . . . . . 16 ((Rel (𝐶 Func (𝐷 ×c 𝐸)) ∧ 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5855, 56, 57sylancr 590 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5917, 4, 58funcf1 17195 . . . . . . . . . . . . . 14 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
6059ffvelrnda 6842 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6261adantr 484 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6359ffvelrnda 6842 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6463adantrl 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6564adantr 484 . . . . . . . . . . 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, 651stf2 17509 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) = (1st ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))))
6766fveq1d 6660 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((1st ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))‘((𝑥(2nd𝑃)𝑦)‘𝑓)))
6858adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
69 simprl 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
70 simprr 772 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
7117, 36, 5, 68, 69, 70funcf2 17197 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))
7271ffvelrnda 6842 . . . . . . . . . 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 6678 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = (1st ‘((𝑥(2nd𝑃)𝑦)‘𝑓)))
746ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
7511ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
7669adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
7770adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
78 simpr 488 . . . . . . . . . . . 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 17518 . . . . . . . . . . 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 6662 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = (1st ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩))
81 fvex 6671 . . . . . . . . . . 11 ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ V
82 fvex 6671 . . . . . . . . . . 11 ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ V
8381, 82op1st 7701 . . . . . . . . . 10 (1st ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd𝐹)𝑦)‘𝑓)
8480, 83eqtrdi 2809 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
8567, 73, 843eqtrd 2797 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
8685mpteq2dva 5127 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓))) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
87 eqid 2758 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
8846adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
894, 5, 87, 88, 61, 64funcf2 17197 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)):(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))⟶(((1st ‘(𝐷 1stF 𝐸))‘((1st𝑃)‘𝑥))(Hom ‘𝐷)((1st ‘(𝐷 1stF 𝐸))‘((1st𝑃)‘𝑦))))
90 fcompt 6886 . . . . . . . 8 (((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)):(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))⟶(((1st ‘(𝐷 1stF 𝐸))‘((1st𝑃)‘𝑥))(Hom ‘𝐷)((1st ‘(𝐷 1stF 𝐸))‘((1st𝑃)‘𝑦))) ∧ (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓))))
9189, 71, 90syl2anc 587 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦))‘((𝑥(2nd𝑃)𝑦)‘𝑓))))
9220adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
9317, 36, 87, 92, 69, 70funcf2 17197 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
9493feqmptd 6721 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
9586, 91, 943eqtr4d 2803 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐹)𝑦))
96953impb 1112 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐹)𝑦))
9796mpoeq3dva 7225 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
9817, 20funcfn2 17198 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
99 fnov 7277 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
10098, 99sylib 221 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
10197, 100eqtr4d 2796 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (2nd𝐹))
10252, 101opeq12d 4771 . 2 (𝜑 → ⟨((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)))⟩ = ⟨(1st𝐹), (2nd𝐹)⟩)
10317, 56, 44cofuval 17211 . 2 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = ⟨((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)))⟩)
104 1st2nd 7742 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
10518, 6, 104sylancr 590 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
106102, 103, 1053eqtr4d 2803 1 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112   × cxp 5522   ↾ cres 5526   ∘ ccom 5528  Rel wrel 5529   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  1st c1st 7691  2nd c2nd 7692  Basecbs 16541  Hom chom 16634  Catccat 16993   Func cfunc 17183   ∘func ccofu 17185   ×c cxpc 17484   1stF c1stf 17485   ⟨,⟩F cprf 17487 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-ixp 8480  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-hom 16647  df-cco 16648  df-cat 16997  df-cid 16998  df-func 17187  df-cofu 17189  df-xpc 17488  df-1stf 17489  df-prf 17491 This theorem is referenced by: (None)
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