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Theorem prf1st 16765
 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 2621 . . . . . . 7 (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2 eqid 2621 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2621 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
41, 2, 3xpcbas 16739 . . . . . . 7 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5 eqid 2621 . . . . . . 7 (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6 prf1st.c . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 16444 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simprd 479 . . . . . . . 8 (𝜑𝐷 ∈ Cat)
109adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11 prf1st.d . . . . . . . . . 10 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
12 funcrcl 16444 . . . . . . . . . 10 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simprd 479 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
1514adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16 eqid 2621 . . . . . . 7 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
17 eqid 2621 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
18 relfunc 16443 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
19 1st2ndbr 7162 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 6, 19sylancr 694 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2117, 2, 20funcf1 16447 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
23 relfunc 16443 . . . . . . . . . . 11 Rel (𝐶 Func 𝐸)
24 1st2ndbr 7162 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2523, 11, 24sylancr 694 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2617, 3, 25funcf1 16447 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
2726ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
28 opelxpi 5108 . . . . . . . 8 ((((1st𝐹)‘𝑥) ∈ (Base‘𝐷) ∧ ((1st𝐺)‘𝑥) ∈ (Base‘𝐸)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
2922, 27, 28syl2anc 692 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
301, 4, 5, 10, 15, 16, 291stf1 16753 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = (1st ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
31 fvex 6158 . . . . . . 7 ((1st𝐹)‘𝑥) ∈ V
32 fvex 6158 . . . . . . 7 ((1st𝐺)‘𝑥) ∈ V
3331, 32op1st 7121 . . . . . 6 (1st ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐹)‘𝑥)
3430, 33syl6eq 2671 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐹)‘𝑥))
3534mpteq2dva 4704 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
36 prf1st.p . . . . . . 7 𝑃 = (𝐹 ⟨,⟩F 𝐺)
37 eqid 2621 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3836, 17, 37, 6, 11prfval 16760 . . . . . 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)‘)⟩))⟩)
39 fvex 6158 . . . . . . . 8 (Base‘𝐶) ∈ V
4039mptex 6440 . . . . . . 7 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
4139, 39mpt2ex 7192 . . . . . . 7 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
4240, 41op1std 7123 . . . . . 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)⟩))
4338, 42syl 17 . . . . 5 (𝜑 → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
44 relfunc 16443 . . . . . . . 8 Rel ((𝐷 ×c 𝐸) Func 𝐷)
451, 9, 14, 161stfcl 16758 . . . . . . . 8 (𝜑 → (𝐷 1stF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐷))
46 1st2ndbr 7162 . . . . . . . 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)))
4744, 45, 46sylancr 694 . . . . . . 7 (𝜑 → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
484, 2, 47funcf1 16447 . . . . . 6 (𝜑 → (1st ‘(𝐷 1stF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐷))
4948feqmptd 6206 . . . . 5 (𝜑 → (1st ‘(𝐷 1stF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 1stF 𝐸))‘𝑢)))
50 fveq2 6148 . . . . 5 (𝑢 = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ → ((1st ‘(𝐷 1stF 𝐸))‘𝑢) = ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
5129, 43, 49, 50fmptco 6351 . . . 4 (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)))
5221feqmptd 6206 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
5335, 51, 523eqtr4d 2665 . . 3 (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)) = (1st𝐹))
549ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
5514ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
56 relfunc 16443 . . . . . . . . . . . . . . . 16 Rel (𝐶 Func (𝐷 ×c 𝐸))
5736, 1, 6, 11prfcl 16764 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
58 1st2ndbr 7162 . . . . . . . . . . . . . . . 16 ((Rel (𝐶 Func (𝐷 ×c 𝐸)) ∧ 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5956, 57, 58sylancr 694 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
6017, 4, 59funcf1 16447 . . . . . . . . . . . . . 14 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
6160ffvelrnda 6315 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6261adantrr 752 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6362adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6460ffvelrnda 6315 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6564adantrl 751 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6665adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
671, 4, 5, 54, 55, 16, 63, 661stf2 16754 . . . . . . . . . 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))))
6867fveq1d 6150 . . . . . . . . 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)))
6959adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
70 simprl 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
71 simprr 795 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
7217, 37, 5, 69, 70, 71funcf2 16449 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))
7372ffvelrnda 6315 . . . . . . . . . 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)))
74 fvres 6164 . . . . . . . . . 10 (((𝑥(2nd𝑃)𝑦)‘𝑓) ∈ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)) → ((1st ↾ (((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = (1st ‘((𝑥(2nd𝑃)𝑦)‘𝑓)))
7573, 74syl 17 . . . . . . . . 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)))
766ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
7711ad2antrr 761 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
7870adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
7971adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
80 simpr 477 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
8136, 17, 37, 76, 77, 78, 79, 80prf2 16763 . . . . . . . . . . 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)⟩)
8281fveq2d 6152 . . . . . . . . . 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)⟩))
83 fvex 6158 . . . . . . . . . . 11 ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ V
84 fvex 6158 . . . . . . . . . . 11 ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ V
8583, 84op1st 7121 . . . . . . . . . 10 (1st ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd𝐹)𝑦)‘𝑓)
8682, 85syl6eq 2671 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((𝑥(2nd𝑃)𝑦)‘𝑓)) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
8768, 75, 863eqtrd 2659 . . . . . . . 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))
8887mpteq2dva 4704 . . . . . . 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)))
89 eqid 2621 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
9047adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
914, 5, 89, 90, 62, 65funcf2 16449 . . . . . . . 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))))
92 fcompt 6354 . . . . . . . 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))))
9391, 72, 92syl2anc 692 . . . . . . 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))))
9420adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
9517, 37, 89, 94, 70, 71funcf2 16449 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
9695feqmptd 6206 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
9788, 93, 963eqtr4d 2665 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐹)𝑦))
98973impb 1257 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐹)𝑦))
9998mpt2eq3dva 6672 . . . 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)))
10017, 20funcfn2 16450 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
101 fnov 6721 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
102100, 101sylib 208 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
10399, 102eqtr4d 2658 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (2nd𝐹))
10453, 103opeq12d 4378 . 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)⟩)
10517, 57, 45cofuval 16463 . 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)))⟩)
106 1st2nd 7159 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
10718, 6, 106sylancr 694 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
108104, 105, 1073eqtr4d 2665 1 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ⟨cop 4154   class class class wbr 4613   ↦ cmpt 4673   × cxp 5072   ↾ cres 5076   ∘ ccom 5078  Rel wrel 5079   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  1st c1st 7111  2nd c2nd 7112  Basecbs 15781  Hom chom 15873  Catccat 16246   Func cfunc 16435   ∘func ccofu 16437   ×c cxpc 16729   1stF c1stf 16730   ⟨,⟩F cprf 16732 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-hom 15887  df-cco 15888  df-cat 16250  df-cid 16251  df-func 16439  df-cofu 16441  df-xpc 16733  df-1stf 16734  df-prf 16736 This theorem is referenced by: (None)
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