Theorem prf1st 18371

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 2761	. . . . . . 7 ⊢ (𝐷 ×c 𝐸) = (𝐷 ×c 𝐸)
2		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		eqid 2761	. . . . . . . 8 ⊢ (Base‘𝐸) = (Base‘𝐸)
4	1, 2, 3	xpcbas 18345	. . . . . . 7 ⊢ ((Base‘𝐷) × (Base‘𝐸)) = (Base‘(𝐷 ×c 𝐸))
5		eqid 2761	. . . . . . 7 ⊢ (Hom ‘(𝐷 ×c 𝐸)) = (Hom ‘(𝐷 ×c 𝐸))
6		prf1st.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		funcrcl 18031	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
8	6, 7	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
9	8	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Cat)
10	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
11		prf1st.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))
12		funcrcl 18031	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Cat)
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
16		eqid 2761	. . . . . . 7 ⊢ (𝐷 1stF 𝐸) = (𝐷 1stF 𝐸)
17		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝐶)
18		relfunc 18030	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
19		1st2ndbr 8051	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
20	18, 6, 19	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
21	17, 2, 20	funcf1 18034	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
22	21	ffvelcdmda 7082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
23		relfunc 18030	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐸)
24		1st2ndbr 8051	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st ‘𝐺)(𝐶 Func 𝐸)(2nd ‘𝐺))
25	23, 11, 24	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐸)(2nd ‘𝐺))
26	17, 3, 25	funcf1 18034	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐸))
27	26	ffvelcdmda 7082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐸))
28	22, 27	opelxpd 5690	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
29	1, 4, 5, 10, 15, 16, 28	1stf1 18359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) = (1st ‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
30		fvex 6896	. . . . . . 7 ⊢ ((1st ‘𝐹)‘𝑥) ∈ V
31		fvex 6896	. . . . . . 7 ⊢ ((1st ‘𝐺)‘𝑥) ∈ V
32	30, 31	op1st 8007	. . . . . 6 ⊢ (1st ‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) = ((1st ‘𝐹)‘𝑥)
33	29, 32	eqtrdi 2812	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) = ((1st ‘𝐹)‘𝑥))
34	33	mpteq2dva 5198	. . . 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 2761	. . . . . . 7 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
37	35, 17, 36, 6, 11	prfval 18366	. . . . . 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 6896	. . . . . . . 8 ⊢ (Base‘𝐶) ∈ V
39	38	mptex 7227	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩) ∈ V
40	38, 38	mpoex 8090	. . . . . . 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 8009	. . . . . 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 18	. . . . 5 ⊢ (𝜑 → (1st ‘𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩))
43		relfunc 18030	. . . . . . . 8 ⊢ Rel ((𝐷 ×c 𝐸) Func 𝐷)
44	1, 9, 14, 16	1stfcl 18364	. . . . . . . 8 ⊢ (𝜑 → (𝐷 1stF 𝐸) ∈ ((𝐷 ×c 𝐸) Func 𝐷))
45		1st2ndbr 8051	. . . . . . . 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 599	. . . . . . 7 ⊢ (𝜑 → (1st ‘(𝐷 1stF 𝐸))((𝐷 ×c 𝐸) Func 𝐷)(2nd ‘(𝐷 1stF 𝐸)))
47	4, 2, 46	funcf1 18034	. . . . . 6 ⊢ (𝜑 → (1st ‘(𝐷 1stF 𝐸)):((Base‘𝐷) × (Base‘𝐸))⟶(Base‘𝐷))
48	47	feqmptd 6951	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐷 1stF 𝐸)) = (𝑢 ∈ ((Base‘𝐷) × (Base‘𝐸)) ↦ ((1st ‘(𝐷 1stF 𝐸))‘𝑢)))
49		fveq2 6883	. . . . 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 7128	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st ‘𝑃)) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘(𝐷 1stF 𝐸))‘⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)))
51	21	feqmptd 6951	. . . 4 ⊢ (𝜑 → (1st ‘𝐹) = (𝑥 ∈ (Base‘𝐶) ↦ ((1st ‘𝐹)‘𝑥)))
52	34, 50, 51	3eqtr4d 2806	. . 3 ⊢ (𝜑 → ((1st ‘(𝐷 1stF 𝐸)) ∘ (1st ‘𝑃)) = (1st ‘𝐹))
53	9	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐷 ∈ Cat)
54	14	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐸 ∈ Cat)
55		relfunc 18030	. . . . . . . . . . . . . . . 16 ⊢ Rel (𝐶 Func (𝐷 ×c 𝐸))
56	35, 1, 6, 11	prfcl 18370	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ (𝐶 Func (𝐷 ×c 𝐸)))
57		1st2ndbr 8051	. . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1st ‘𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd ‘𝑃))
59	17, 4, 58	funcf1 18034	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1st ‘𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
60	59	ffvelcdmda 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1st ‘𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
61	60	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
62	61	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((1st ‘𝑃)‘𝑥) ∈ ((Base‘𝐷) × (Base‘𝐸)))
63	59	ffvelcdmda 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
64	63	adantrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st ‘𝑃)‘𝑦) ∈ ((Base‘𝐷) × (Base‘𝐸)))
65	64	adantr 486	. . . . . . . . . . 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 18360	. . . . . . . . . 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 6885	. . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝑃)(𝐶 Func (𝐷 ×c 𝐸))(2nd ‘𝑃))
69		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
70		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
71	17, 36, 5, 68, 69, 70	funcf2 18036	. . . . . . . . . . 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 7082	. . . . . . . . . 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 6903	. . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐹 ∈ (𝐶 Func 𝐷))
75	11	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝐺 ∈ (𝐶 Func 𝐸))
76	69	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑥 ∈ (Base‘𝐶))
77	70	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) → 𝑦 ∈ (Base‘𝐶))
78		simpr 490	. . . . . . . . . . . 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 18369	. . . . . . . . . . 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 6887	. . . . . . . . . 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 6896	. . . . . . . . . . 11 ⊢ ((𝑥(2nd ‘𝐹)𝑦)‘𝑓) ∈ V
82		fvex 6896	. . . . . . . . . . 11 ⊢ ((𝑥(2nd ‘𝐺)𝑦)‘𝑓) ∈ V
83	81, 82	op1st 8007	. . . . . . . . . 10 ⊢ (1st ‘⟨((𝑥(2nd ‘𝐹)𝑦)‘𝑓), ((𝑥(2nd ‘𝐺)𝑦)‘𝑓)⟩) = ((𝑥(2nd ‘𝐹)𝑦)‘𝑓)
84	80, 83	eqtrdi 2812	. . . . . . . . 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 2800	. . . . . . . 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 5198	. . . . . . 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 2761	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
88	46	adantr 486	. . . . . . . . 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 18036	. . . . . . . 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 7132	. . . . . . . 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 596	. . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
93	17, 36, 87, 92, 69, 70	funcf2 18036	. . . . . . . 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 6951	. . . . . . 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 2806	. . . . . 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 1132	. . . . 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 7495	. . . 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 18037	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)))
99		fnov 7549	. . . . 5 ⊢ ((2nd ‘𝐹) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
100	98, 99	sylib 221	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ (𝑥(2nd ‘𝐹)𝑦)))
101	97, 100	eqtr4d 2799	. . 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 4841	. 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 18050	. 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 8048	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
105	18, 6, 104	sylancr 599	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
106	102, 103, 105	3eqtr4d 2806	1 ⊢ (𝜑 → ((𝐷 1stF 𝐸) ∘func 𝑃) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649   ↾ cres 5653   ∘ ccom 5655  Rel wrel 5656   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1^st c1st 7997  2^nd c2nd 7998  Basecbs 17380  Hom chom 17432  Catccat 17831   Func cfunc 18022   ∘_func ccofu 18024   ×_c cxpc 18335   1^st_F c1stf 18336   ⟨,⟩_F cprf 18338

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-cat 17835  df-cid 17836  df-func 18026  df-cofu 18028  df-xpc 18339  df-1stf 18340  df-prf 18342

This theorem is used by: (None)