Theorem prf1st 18165

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	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
prf1st.d	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))

Assertion

Ref	Expression
prf1st	⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)

Proof of Theorem prf1st

Dummy variables 𝑓 ℎ 𝑥 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . . . . 7 ⊢ (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
4	1, 2, 3	xpcbas 18139	. . . . . . 7 ⊢ ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5		eqid 2729	. . . . . . 7 ⊢ (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6		prf1st.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 17825	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9	8	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11		prf1st.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
12		funcrcl 17825	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16		eqid 2729	. . . . . . 7 ⊢ (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
17		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		relfunc 17824	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
19		1st2ndbr 8021	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
20	18, 6, 19	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
21	17, 2, 20	funcf1 17828	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
22	21	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
23		relfunc 17824	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐸)
24		1st2ndbr 8021	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st ‘𝐺)(𝐶 Func 𝐸)(2nd ‘𝐺))
25	23, 11, 24	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐸)(2nd ‘𝐺))
26	17, 3, 25	funcf1 17828	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐸))
27	26	ffvelcdmda 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐸))
28	22, 27	opelxpd 5677	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
29	1, 4, 5, 10, 15, 16, 28	1stf1 18153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) = (1st ‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
30		fvex 6871	. . . . . . 7 ⊢ ((1st ‘𝐹)‘𝑥) ∈ V
31		fvex 6871	. . . . . . 7 ⊢ ((1st ‘𝐺)‘𝑥) ∈ V
32	30, 31	op1st 7976	. . . . . 6 ⊢ (1st ‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) = ((1st ‘𝐹)‘𝑥)
33	29, 32	eqtrdi 2780	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) = ((1st ‘𝐹)‘𝑥))
34	33	mpteq2dva 5200	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)))
35		prf1st.p	. . . . . . 7 ⊢ 𝑃 = (𝐹 ⟨,⟩F 𝐺)
36		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	35, 17, 36, 6, 11	prfval 18160	. . . . . 6 ⊢ (𝜑 → 𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (ℎ ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩)
38		fvex 6871	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
39	38	mptex 7197	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) ∈ V
40	38, 38	mpoex 8058	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (ℎ ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩)) ∈ V
41	39, 40	op1std 7978	. . . . . 6 ⊢ (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (ℎ ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd ‘𝐹)𝑦)‘ℎ), ((𝑥(2nd ‘𝐺)𝑦)‘ℎ)⟩))⟩ → (1st ‘𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
42	37, 41	syl 17	. . . . 5 ⊢ (𝜑 → (1st ‘𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
43		relfunc 17824	. . . . . . . 8 ⊢ Rel ((𝐷 ×c 𝐸) Func 𝐷)
44	1, 9, 14, 16	1stfcl 18158	. . . . . . . 8 ⊢ (𝜑 → (𝐷 1stF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐷))
45		1st2ndbr 8021	. . . . . . . 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)))
46	43, 44, 45	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
47	4, 2, 46	funcf1 17828	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝐷 1stF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐷))
48	47	feqmptd 6929	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐷 1stF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 1stF 𝐸))‘𝑢)))
49		fveq2 6858	. . . . 5 ⊢ (𝑢 = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ → ((1st ‘(𝐷 1stF 𝐸))‘𝑢) = ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
50	28, 42, 48, 49	fmptco 7101	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st ‘𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)))
51	21	feqmptd 6929	. . . 4 ⊢ (𝜑 → (1st ‘𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)))
52	34, 50, 51	3eqtr4d 2774	. . 3 ⊢ (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st ‘𝑃)) = (1st ‘𝐹))
53	9	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
54	14	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
55		relfunc 17824	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝐶 Func (𝐷 ×c 𝐸))
56	35, 1, 6, 11	prfcl 18164	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
57		1st2ndbr 8021	. . . . . . . . . . . . . . . 16 ⊢ ((Rel (𝐶 Func (𝐷 ×c 𝐸)) ∧ 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸))) → (1st ‘𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd ‘𝑃))
58	55, 56, 57	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1st ‘𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd ‘𝑃))
59	17, 4, 58	funcf1 17828	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1st ‘𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
60	59	ffvelcdmda 7056	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
61	60	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
62	61	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
63	59	ffvelcdmda 7056	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
64	63	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
65	64	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
66	1, 4, 5, 53, 54, 16, 62, 65	1stf2 18154	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) = (1st ↾ (((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦))))
67	66	fveq1d 6860	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)) = ((1st ↾ (((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦)))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)))
68	58	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd ‘𝑃))
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))
71	17, 36, 5, 68, 69, 70	funcf2 17830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦)))
72	71	ffvelcdmda 7056	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝑃)𝑦)‘𝑓) ∈ (((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦)))
73	72	fvresd 6878	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ↾ (((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦)))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)) = (1st ‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)))
74	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
75	11	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
76	69	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
77	70	adantr 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))
79	35, 17, 36, 74, 75, 76, 77, 78	prf2 18163	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd ‘𝑃)𝑦)‘𝑓) = ⟨((𝑥(2nd ‘𝐹)𝑦)‘𝑓), ((𝑥(2nd ‘𝐺)𝑦)‘𝑓)⟩)
80	79	fveq2d 6862	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)) = (1st ‘⟨((𝑥(2nd ‘𝐹)𝑦)‘𝑓), ((𝑥(2nd ‘𝐺)𝑦)‘𝑓)⟩))
81		fvex 6871	. . . . . . . . . . 11 ⊢ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ V
82		fvex 6871	. . . . . . . . . . 11 ⊢ ((𝑥(2nd ‘𝐺)𝑦)‘𝑓) ∈ V
83	81, 82	op1st 7976	. . . . . . . . . 10 ⊢ (1st ‘⟨((𝑥(2nd ‘𝐹)𝑦)‘𝑓), ((𝑥(2nd ‘𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)
84	80, 83	eqtrdi 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → (1st ‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓))
85	67, 73, 84	3eqtrd 2768	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓)) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓))
86	85	mpteq2dva 5200	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓))) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
87		eqid 2729	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
88	46	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
89	4, 5, 87, 88, 61, 64	funcf2 17830	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)):(((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦))⟶(((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝑃)‘𝑥))(Hom ‘𝐷)((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝑃)‘𝑦))))
90		fcompt 7105	. . . . . . . 8 ⊢ (((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)):(((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦))⟶(((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝑃)‘𝑥))(Hom ‘𝐷)((1st ‘(𝐷 1stF 𝐸))‘((1st ‘𝑃)‘𝑦))) ∧ (𝑥(2nd ‘𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝑃)‘𝑥)(Hom ‘(𝐷 ×c 𝐸))((1st ‘𝑃)‘𝑦))) → ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓))))
91	89, 71, 90	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦)) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦))‘((𝑥(2nd ‘𝑃)𝑦)‘𝑓))))
92	20	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
93	17, 36, 87, 92, 69, 70	funcf2 17830	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1st ‘𝐹)‘𝑦)))
94	93	feqmptd 6929	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd ‘𝐹)𝑦) = (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)))
95	86, 91, 94	3eqtr4d 2774	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
96	95	3impb 1114	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦)) = (𝑥(2nd ‘𝐹)𝑦))
97	96	mpoeq3dva 7466	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
98	17, 20	funcfn2 17831	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
99		fnov 7520	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
100	98, 99	sylib 218	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
101	97, 100	eqtr4d 2767	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦))) = (2nd ‘𝐹))
102	52, 101	opeq12d 4845	. 2 ⊢ (𝜑 → ⟨((1st ‘(𝐷 1stF 𝐸)) ∘ (1st ‘𝑃)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦)))⟩ = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
103	17, 56, 44	cofuval 17844	. 2 ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = ⟨((1st ‘(𝐷 1stF 𝐸)) ∘ (1st ‘𝑃)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑃)‘𝑥)(2nd ‘(𝐷 1stF 𝐸))((1st ‘𝑃)‘𝑦)) ∘ (𝑥(2nd ‘𝑃)𝑦)))⟩)
104		1st2nd 8018	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
105	18, 6, 104	sylancr 587	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
106	102, 103, 105	3eqtr4d 2774	1 ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636   ↾ cres 5640   ∘ ccom 5642  Rel wrel 5643   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  Hom chom 17231  Catccat 17625   Func cfunc 17816   ∘_func ccofu 17818   ×_c cxpc 18129   1^st_F c1stf 18130   ⟨,⟩_F cprf 18132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-cat 17629  df-cid 17630  df-func 17820  df-cofu 17822  df-xpc 18133  df-1stf 18134  df-prf 18136

This theorem is referenced by: (None)