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Theorem prf2nd 18201
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 2727 . . . . . . 7 (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2 eqid 2727 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2727 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
41, 2, 3xpcbas 18174 . . . . . . 7 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5 eqid 2727 . . . . . . 7 (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6 prf1st.c . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 17854 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simprd 494 . . . . . . . 8 (𝜑𝐷 ∈ Cat)
109adantr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11 prf1st.d . . . . . . . . . 10 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
12 funcrcl 17854 . . . . . . . . . 10 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simprd 494 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
1514adantr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16 eqid 2727 . . . . . . 7 (𝐷 2ndF 𝐸) = (𝐷 2ndF 𝐸)
17 eqid 2727 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
18 relfunc 17853 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
19 1st2ndbr 8050 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 6, 19sylancr 585 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2117, 2, 20funcf1 17857 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelcdmda 7097 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
23 relfunc 17853 . . . . . . . . . . 11 Rel (𝐶 Func 𝐸)
24 1st2ndbr 8050 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2523, 11, 24sylancr 585 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2617, 3, 25funcf1 17857 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
2726ffvelcdmda 7097 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
2822, 27opelxpd 5719 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
291, 4, 5, 10, 15, 16, 282ndf1 18191 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = (2nd ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
30 fvex 6913 . . . . . . 7 ((1st𝐹)‘𝑥) ∈ V
31 fvex 6913 . . . . . . 7 ((1st𝐺)‘𝑥) ∈ V
3230, 31op2nd 8006 . . . . . 6 (2nd ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐺)‘𝑥)
3329, 32eqtrdi 2783 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐺)‘𝑥))
3433mpteq2dva 5250 . . . 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 2727 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3735, 17, 36, 6, 11prfval 18195 . . . . . 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 6913 . . . . . . . 8 (Base‘𝐶) ∈ V
3938mptex 7239 . . . . . . 7 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
4038, 38mpoex 8088 . . . . . . 7 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
4139, 40op1std 8007 . . . . . 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 17853 . . . . . . . 8 Rel ((𝐷 ×c 𝐸) Func 𝐸)
441, 9, 14, 162ndfcl 18194 . . . . . . . 8 (𝜑 → (𝐷 2ndF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐸))
45 1st2ndbr 8050 . . . . . . . 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 585 . . . . . . 7 (𝜑 → (1st ‘(𝐷 2ndF 𝐸))((𝐷 ×c 𝐸) Func 𝐸)(2nd ‘(𝐷 2ndF 𝐸)))
474, 3, 46funcf1 17857 . . . . . 6 (𝜑 → (1st ‘(𝐷 2ndF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐸))
4847feqmptd 6970 . . . . 5 (𝜑 → (1st ‘(𝐷 2ndF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 2ndF 𝐸))‘𝑢)))
49 fveq2 6900 . . . . 5 (𝑢 = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ → ((1st ‘(𝐷 2ndF 𝐸))‘𝑢) = ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
5028, 42, 48, 49fmptco 7142 . . . 4 (𝜑 → ((1st ‘(𝐷 2ndF 𝐸)) ∘ (1st𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 2ndF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)))
5126feqmptd 6970 . . . 4 (𝜑 → (1st𝐺) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐺)‘𝑥)))
5234, 50, 513eqtr4d 2777 . . 3 (𝜑 → ((1st ‘(𝐷 2ndF 𝐸)) ∘ (1st𝑃)) = (1st𝐺))
539ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
5414ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
55 relfunc 17853 . . . . . . . . . . . . . . . 16 Rel (𝐶 Func (𝐷 ×c 𝐸))
5635, 1, 6, 11prfcl 18199 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
57 1st2ndbr 8050 . . . . . . . . . . . . . . . 16 ((Rel (𝐶 Func (𝐷 ×c 𝐸)) ∧ 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5855, 56, 57sylancr 585 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5917, 4, 58funcf1 17857 . . . . . . . . . . . . . 14 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
6059ffvelcdmda 7097 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6160adantrr 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6261adantr 479 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6359ffvelcdmda 7097 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6463adantrl 714 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6564adantr 479 . . . . . . . . . . 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 18192 . . . . . . . . . 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 6902 . . . . . . . . 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 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
69 simprl 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
70 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
7117, 36, 5, 68, 69, 70funcf2 17859 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))
7271ffvelcdmda 7097 . . . . . . . . . 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 6920 . . . . . . . . 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 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
7511ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
7669adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
7770adantr 479 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
78 simpr 483 . . . . . . . . . . . 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 18198 . . . . . . . . . . 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 6904 . . . . . . . . . 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 6913 . . . . . . . . . . 11 ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ V
82 fvex 6913 . . . . . . . . . . 11 ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ V
8381, 82op2nd 8006 . . . . . . . . . 10 (2nd ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd𝐺)𝑦)‘𝑓)
8480, 83eqtrdi 2783 . . . . . . . . 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 2771 . . . . . . . 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 5250 . . . . . . 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 2727 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
8846adantr 479 . . . . . . . . 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 17859 . . . . . . . 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 7146 . . . . . . . 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 582 . . . . . . 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 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
9317, 36, 87, 92, 69, 70funcf2 17859 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
9493feqmptd 6970 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐺)𝑦)‘𝑓)))
9586, 91, 943eqtr4d 2777 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐺)𝑦))
96953impb 1112 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐺)𝑦))
9796mpoeq3dva 7501 . . . 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 17860 . . . . 5 (𝜑 → (2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)))
99 fnov 7556 . . . . 5 ((2nd𝐺) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
10098, 99sylib 217 . . . 4 (𝜑 → (2nd𝐺) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐺)𝑦)))
10197, 100eqtr4d 2770 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 2ndF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (2nd𝐺))
10252, 101opeq12d 4884 . 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 17873 . 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 8047 . . 3 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → 𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
10523, 11, 104sylancr 585 . 2 (𝜑𝐺 = ⟨(1st𝐺), (2nd𝐺)⟩)
106102, 103, 1053eqtr4d 2777 1 (𝜑 → ((𝐷 2ndF 𝐸) ∘func 𝑃) = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4636   class class class wbr 5150  cmpt 5233   × cxp 5678  cres 5682  ccom 5684  Rel wrel 5685   Fn wfn 6546  wf 6547  cfv 6551  (class class class)co 7424  cmpo 7426  1st c1st 7995  2nd c2nd 7996  Basecbs 17185  Hom chom 17249  Catccat 17649   Func cfunc 17845  func ccofu 17847   ×c cxpc 18164   2ndF c2ndf 18166   ⟨,⟩F cprf 18167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-map 8851  df-ixp 8921  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12509  df-z 12595  df-dec 12714  df-uz 12859  df-fz 13523  df-struct 17121  df-slot 17156  df-ndx 17168  df-base 17186  df-hom 17262  df-cco 17263  df-cat 17653  df-cid 17654  df-func 17849  df-cofu 17851  df-xpc 18168  df-2ndf 18170  df-prf 18171
This theorem is referenced by: (None)
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