Theorem cvmlift3lem5 32572

Description: Lemma for cvmlift2 32565. (Contributed by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
cvmlift3.h	⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))

Assertion

Ref	Expression
cvmlift3lem5	⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)

Distinct variable groups:   𝑧,𝑓,𝑔,𝑥   𝑓,𝐽   𝑥,𝑔,𝐽   𝑓,𝐹,𝑔   𝑥,𝑧,𝐹   𝑓,𝐻,𝑔,𝑥,𝑧   𝐵,𝑓,𝑔,𝑥,𝑧   𝑓,𝐺,𝑔,𝑥,𝑧   𝐶,𝑓,𝑔,𝑥,𝑧   𝜑,𝑓,𝑥   𝑓,𝐾,𝑔,𝑥,𝑧   𝑃,𝑓,𝑔,𝑥,𝑧   𝑓,𝑂,𝑔,𝑥,𝑧   𝑓,𝑌,𝑔,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧,𝑔)   𝐽(𝑧)

Proof of Theorem cvmlift3lem5

Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . . . 5 ⊢ (𝐻‘𝑦) = (𝐻‘𝑦)
2		cvmlift3.b	. . . . . 6 ⊢ 𝐵 = ∪ 𝐶
3		cvmlift3.y	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
4		cvmlift3.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
5		cvmlift3.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ SConn)
6		cvmlift3.l	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
7		cvmlift3.o	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
8		cvmlift3.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
9		cvmlift3.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
10		cvmlift3.e	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
11		cvmlift3.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
12	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem4 32571	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ((𝐻‘𝑦) = (𝐻‘𝑦) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦))))
13	1, 12	mpbii 235	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦)))
14		df-3an 1085	. . . . . 6 ⊢ (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦)) ↔ (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦)))
15		eqid 2823	. . . . . . . . . . . 12 ⊢ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))
16	4	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓 ∈ (II Cn 𝐾))
18	8	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19		cnco 21876	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
20	17, 18, 19	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
21	9	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑃 ∈ 𝐵)
22		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘0) = 𝑂)
23	22	fveq2d 6676	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘0)) = (𝐺‘𝑂))
24		iiuni 23491	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) = ∪ II
25	24, 3	cnf 21856	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶𝑌)
26	17, 25	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓:(0[,]1)⟶𝑌)
27		0elunit 12858	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0[,]1)
28		fvco3 6762	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
29	26, 27, 28	sylancl 588	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
30	10	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹‘𝑃) = (𝐺‘𝑂))
31	23, 29, 30	3eqtr4rd 2869	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹‘𝑃) = ((𝐺 ∘ 𝑓)‘0))
32	2, 15, 16, 20, 21, 31	cvmliftiota 32550	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) ∧ (𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺 ∘ 𝑓) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘0) = 𝑃))
33	32	simp2d 1139	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))) = (𝐺 ∘ 𝑓))
34	33	fveq1d 6674	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))‘1) = ((𝐺 ∘ 𝑓)‘1))
35	32	simp1d 1138	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶))
36	24, 2	cnf 21856	. . . . . . . . . . 11 ⊢ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) ∈ (II Cn 𝐶) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
37	35, 36	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵)
38		1elunit 12859	. . . . . . . . . 10 ⊢ 1 ∈ (0[,]1)
39		fvco3 6762	. . . . . . . . . 10 ⊢ (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)):(0[,]1)⟶𝐵 ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))‘1) = (𝐹‘((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1)))
40	37, 38, 39	sylancl 588	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐹 ∘ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))‘1) = (𝐹‘((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1)))
41		fvco3 6762	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶𝑌 ∧ 1 ∈ (0[,]1)) → ((𝐺 ∘ 𝑓)‘1) = (𝐺‘(𝑓‘1)))
42	26, 38, 41	sylancl 588	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺 ∘ 𝑓)‘1) = (𝐺‘(𝑓‘1)))
43		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘1) = 𝑦)
44	43	fveq2d 6676	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘1)) = (𝐺‘𝑦))
45	42, 44	eqtrd 2858	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺 ∘ 𝑓)‘1) = (𝐺‘𝑦))
46	34, 40, 45	3eqtr3d 2866	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹‘((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1)) = (𝐺‘𝑦))
47		fveqeq2 6681	. . . . . . . 8 ⊢ (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦) → ((𝐹‘((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1)) = (𝐺‘𝑦) ↔ (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦)))
48	46, 47	syl5ibcom 247	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦) → (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦)))
49	48	expimpd 456	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → ((((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦) ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦)) → (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦)))
50	14, 49	syl5bi 244	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦)) → (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦)))
51	50	rexlimdva 3286	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐻‘𝑦)) → (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦)))
52	13, 51	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦))
53	52	mpteq2dva 5163	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝐹‘(𝐻‘𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)))
54	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem3 32570	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
55	54	ffvelrnda 6853	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻‘𝑦) ∈ 𝐵)
56	54	feqmptd 6735	. . 3 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ 𝑌 ↦ (𝐻‘𝑦)))
57		cvmcn 32511	. . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
58		eqid 2823	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
59	2, 58	cnf 21856	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
60	4, 57, 59	3syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
61	60	feqmptd 6735	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐵 ↦ (𝐹‘𝑤)))
62		fveq2 6672	. . 3 ⊢ (𝑤 = (𝐻‘𝑦) → (𝐹‘𝑤) = (𝐹‘(𝐻‘𝑦)))
63	55, 56, 61, 62	fmptco 6893	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = (𝑦 ∈ 𝑌 ↦ (𝐹‘(𝐻‘𝑦))))
64	3, 58	cnf 21856	. . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
65	8, 64	syl 17	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝐽)
66	65	feqmptd 6735	. 2 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)))
67	53, 63, 66	3eqtr4d 2868	1 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  ∪ cuni 4840   ↦ cmpt 5148   ∘ ccom 5561  ⟶wf 6353  ‘cfv 6357  ℩crio 7115  (class class class)co 7158  0cc0 10539  1c1 10540  [,]cicc 12744   Cn ccn 21834  𝑛-Locally cnlly 22075  IIcii 23485  PConncpconn 32468  SConncsconn 32469   CovMap ccvm 32504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-addf 10618  ax-mulf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-ec 8293  df-map 8410  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-rest 16698  df-topn 16699  df-0g 16717  df-gsum 16718  df-topgen 16719  df-pt 16720  df-prds 16723  df-xrs 16777  df-qtop 16782  df-imas 16783  df-xps 16785  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-submnd 17959  df-mulg 18227  df-cntz 18449  df-cmn 18910  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-mopn 20543  df-cnfld 20548  df-top 21504  df-topon 21521  df-topsp 21543  df-bases 21556  df-cld 21629  df-ntr 21630  df-cls 21631  df-nei 21708  df-cn 21837  df-cnp 21838  df-cmp 21997  df-conn 22022  df-lly 22076  df-nlly 22077  df-tx 22172  df-hmeo 22365  df-xms 22932  df-ms 22933  df-tms 22934  df-ii 23487  df-htpy 23576  df-phtpy 23577  df-phtpc 23598  df-pco 23611  df-pconn 32470  df-sconn 32471  df-cvm 32505

This theorem is referenced by:  cvmlift3lem6  32573  cvmlift3lem7  32574  cvmlift3lem9  32576