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Theorem prf1st 17110
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 2765 . . . . . . 7 (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2 eqid 2765 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
3 eqid 2765 . . . . . . . 8 (Base‘𝐸) = (Base‘𝐸)
41, 2, 3xpcbas 17084 . . . . . . 7 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5 eqid 2765 . . . . . . 7 (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6 prf1st.c . . . . . . . . . 10 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 16788 . . . . . . . . . 10 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simprd 489 . . . . . . . 8 (𝜑𝐷 ∈ Cat)
109adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11 prf1st.d . . . . . . . . . 10 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
12 funcrcl 16788 . . . . . . . . . 10 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . . . . . 9 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simprd 489 . . . . . . . 8 (𝜑𝐸 ∈ Cat)
1514adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16 eqid 2765 . . . . . . 7 (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
17 eqid 2765 . . . . . . . . . 10 (Base‘𝐶) = (Base‘𝐶)
18 relfunc 16787 . . . . . . . . . . 11 Rel (𝐶 Func 𝐷)
19 1st2ndbr 7417 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2018, 6, 19sylancr 581 . . . . . . . . . 10 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2117, 2, 20funcf1 16791 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2221ffvelrnda 6549 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
23 relfunc 16787 . . . . . . . . . . 11 Rel (𝐶 Func 𝐸)
24 1st2ndbr 7417 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2523, 11, 24sylancr 581 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
2617, 3, 25funcf1 16791 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
2726ffvelrnda 6549 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
28 opelxpi 5314 . . . . . . . 8 ((((1st𝐹)‘𝑥) ∈ (Base‘𝐷) ∧ ((1st𝐺)‘𝑥) ∈ (Base‘𝐸)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
2922, 27, 28syl2anc 579 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
301, 4, 5, 10, 15, 16, 291stf1 17098 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = (1st ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
31 fvex 6388 . . . . . . 7 ((1st𝐹)‘𝑥) ∈ V
32 fvex 6388 . . . . . . 7 ((1st𝐺)‘𝑥) ∈ V
3331, 32op1st 7374 . . . . . 6 (1st ‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐹)‘𝑥)
3430, 33syl6eq 2815 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ((1st𝐹)‘𝑥))
3534mpteq2dva 4903 . . . 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 2765 . . . . . . 7 (Hom ‘𝐶) = (Hom ‘𝐶)
3836, 17, 37, 6, 11prfval 17105 . . . . . 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 6388 . . . . . . . 8 (Base‘𝐶) ∈ V
4039mptex 6679 . . . . . . 7 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
4139, 39mpt2ex 7448 . . . . . . 7 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
4240, 41op1std 7376 . . . . . 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 16787 . . . . . . . 8 Rel ((𝐷 ×c 𝐸) Func 𝐷)
451, 9, 14, 161stfcl 17103 . . . . . . . 8 (𝜑 → (𝐷 1stF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐷))
46 1st2ndbr 7417 . . . . . . . 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 581 . . . . . . 7 (𝜑 → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
484, 2, 47funcf1 16791 . . . . . 6 (𝜑 → (1st ‘(𝐷 1stF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐷))
4948feqmptd 6438 . . . . 5 (𝜑 → (1st ‘(𝐷 1stF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 1stF 𝐸))‘𝑢)))
50 fveq2 6375 . . . . 5 (𝑢 = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ → ((1st ‘(𝐷 1stF 𝐸))‘𝑢) = ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
5129, 43, 49, 50fmptco 6587 . . . 4 (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)))
5221feqmptd 6438 . . . 4 (𝜑 → (1st𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st𝐹)‘𝑥)))
5335, 51, 523eqtr4d 2809 . . 3 (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st𝑃)) = (1st𝐹))
549ad2antrr 717 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
5514ad2antrr 717 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
56 relfunc 16787 . . . . . . . . . . . . . . . 16 Rel (𝐶 Func (𝐷 ×c 𝐸))
5736, 1, 6, 11prfcl 17109 . . . . . . . . . . . . . . . 16 (𝜑𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
58 1st2ndbr 7417 . . . . . . . . . . . . . . . 16 ((Rel (𝐶 Func (𝐷 ×c 𝐸)) ∧ 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
5956, 57, 58sylancr 581 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
6017, 4, 59funcf1 16791 . . . . . . . . . . . . . 14 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
6160ffvelrnda 6549 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6261adantrr 708 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6362adantr 472 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6460ffvelrnda 6549 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6564adantrl 707 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
6665adantr 472 . . . . . . . . . . 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 17099 . . . . . . . . . 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 6377 . . . . . . . . 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 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd𝑃))
70 simprl 787 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
71 simprr 789 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
7217, 37, 5, 69, 70, 71funcf2 16793 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st𝑃)‘𝑦)))
7372ffvelrnda 6549 . . . . . . . . . 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 6394 . . . . . . . . . 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 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
7711ad2antrr 717 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
7870adantr 472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
7971adantr 472 . . . . . . . . . . . 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 17108 . . . . . . . . . . 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 6379 . . . . . . . . . 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 6388 . . . . . . . . . . 11 ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ V
84 fvex 6388 . . . . . . . . . . 11 ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ V
8583, 84op1st 7374 . . . . . . . . . 10 (1st ‘⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd𝐹)𝑦)‘𝑓)
8682, 85syl6eq 2815 . . . . . . . . 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 2803 . . . . . . . 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 4903 . . . . . . 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 2765 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
9047adantr 472 . . . . . . . . 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 16793 . . . . . . . 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 6591 . . . . . . . 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 579 . . . . . . 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 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
9517, 37, 89, 94, 70, 71funcf2 16793 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
9695feqmptd 6438 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd𝐹)𝑦)‘𝑓)))
9788, 93, 963eqtr4d 2809 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐹)𝑦))
98973impb 1143 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦)) = (𝑥(2nd𝐹)𝑦))
9998mpt2eq3dva 6917 . . . 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 16794 . . . . 5 (𝜑 → (2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
101 fnov 6966 . . . . 5 ((2nd𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
102100, 101sylib 209 . . . 4 (𝜑 → (2nd𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd𝐹)𝑦)))
10399, 102eqtr4d 2802 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st𝑃)‘𝑦)) ∘ (𝑥(2nd𝑃)𝑦))) = (2nd𝐹))
10453, 103opeq12d 4567 . 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 16807 . 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 7414 . . 3 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
10718, 6, 106sylancr 581 . 2 (𝜑𝐹 = ⟨(1st𝐹), (2nd𝐹)⟩)
108104, 105, 1073eqtr4d 2809 1 (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  cop 4340   class class class wbr 4809  cmpt 4888   × cxp 5275  cres 5279  ccom 5281  Rel wrel 5282   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  1st c1st 7364  2nd c2nd 7365  Basecbs 16130  Hom chom 16225  Catccat 16590   Func cfunc 16779  func ccofu 16781   ×c cxpc 17074   1stF c1stf 17075   ⟨,⟩F cprf 17077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-hom 16238  df-cco 16239  df-cat 16594  df-cid 16595  df-func 16783  df-cofu 16785  df-xpc 17078  df-1stf 17079  df-prf 17081
This theorem is referenced by: (None)
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