Theorem cvmlift3lem5 36174

Description: Lemma for cvmlift2 36167. (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 2760	. . . . 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 36173	. . . . 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 236	. . . 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 1105	. . . . . 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 2760	. . . . . . . . . . . 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 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐹 ∈ (𝐶 CovMap 𝐽))
17		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓 ∈ (II Cn 𝐾))
18	8	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝐺 ∈ (𝐾 Cn 𝐽))
19		cnco 23589	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
20	17, 18, 19	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺 ∘ 𝑓) ∈ (II Cn 𝐽))
21	9	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑃 ∈ 𝐵)
22		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘0) = 𝑂)
23	22	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘0)) = (𝐺‘𝑂))
24		iiuni 25207	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) = ∪ II
25	24, 3	cnf 23569	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (II Cn 𝐾) → 𝑓:(0[,]1)⟶𝑌)
26	17, 25	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → 𝑓:(0[,]1)⟶𝑌)
27		0elunit 13603	. . . . . . . . . . . . . 14 ⊢ 0 ∈ (0[,]1)
28		fvco3 6985	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
29	26, 27, 28	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺 ∘ 𝑓)‘0) = (𝐺‘(𝑓‘0)))
30	10	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐹‘𝑃) = (𝐺‘𝑂))
31	23, 29, 30	3eqtr4rd 2806	. . . . . . . . . . . 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 36152	. . . . . . . . . . 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 1161	. . . . . . . . . 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 6887	. . . . . . . . 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 1160	. . . . . . . . . . 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 23569	. . . . . . . . . . 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 18	. . . . . . . . . 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 13604	. . . . . . . . . 10 ⊢ 1 ∈ (0[,]1)
39		fvco3 6985	. . . . . . . . . 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 598	. . . . . . . . 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 6985	. . . . . . . . . . 11 ⊢ ((𝑓:(0[,]1)⟶𝑌 ∧ 1 ∈ (0[,]1)) → ((𝐺 ∘ 𝑓)‘1) = (𝐺‘(𝑓‘1)))
42	26, 38, 41	sylancl 598	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺 ∘ 𝑓)‘1) = (𝐺‘(𝑓‘1)))
43		simprr 785	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝑓‘1) = 𝑦)
44	43	fveq2d 6889	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → (𝐺‘(𝑓‘1)) = (𝐺‘𝑦))
45	42, 44	eqtrd 2795	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝑌) ∧ 𝑓 ∈ (II Cn 𝐾)) ∧ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑦)) → ((𝐺 ∘ 𝑓)‘1) = (𝐺‘𝑦))
46	34, 40, 45	3eqtr3d 2803	. . . . . . . 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 6894	. . . . . . . 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 248	. . . . . . 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 459	. . . . . 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	biimtrid 245	. . . . 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 3163	. . . 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 16	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(𝐻‘𝑦)) = (𝐺‘𝑦))
53	52	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝐹‘(𝐻‘𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)))
54	2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cvmlift3lem3 36172	. . . 4 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
55	54	ffvelcdmda 7084	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐻‘𝑦) ∈ 𝐵)
56	54	feqmptd 6953	. . 3 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ 𝑌 ↦ (𝐻‘𝑦)))
57		cvmcn 36113	. . . . 5 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
58		eqid 2760	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
59	2, 58	cnf 23569	. . . . 5 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
60	4, 57, 59	3syl 19	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
61	60	feqmptd 6953	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐵 ↦ (𝐹‘𝑤)))
62		fveq2 6885	. . 3 ⊢ (𝑤 = (𝐻‘𝑦) → (𝐹‘𝑤) = (𝐹‘(𝐻‘𝑦)))
63	55, 56, 61, 62	fmptco 7130	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = (𝑦 ∈ 𝑌 ↦ (𝐹‘(𝐻‘𝑦))))
64	3, 58	cnf 23569	. . . 4 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
65	8, 64	syl 18	. . 3 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝐽)
66	65	feqmptd 6953	. 2 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)))
67	53, 63, 66	3eqtr4d 2805	1 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ∪ cuni 4866   ↦ cmpt 5185   ∘ ccom 5654  ⟶wf 6534  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  0cc0 11203  1c1 11204  [,]cicc 13482   Cn ccn 23547  𝑛-Locally cnlly 23789  IIcii 25201  PConncpconn 36070  SConncsconn 36071   CovMap ccvm 36106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281  ax-addf 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8180  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-er 8720  df-ec 8722  df-map 8852  df-ixp 8929  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-fsupp 9357  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9507  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-q 13079  df-rp 13124  df-xneg 13244  df-xadd 13245  df-xmul 13246  df-ioo 13483  df-ico 13485  df-icc 13486  df-fz 13643  df-fzo 13792  df-fl 13935  df-seq 14148  df-exp 14208  df-hash 14478  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-clim 15660  df-sum 15859  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-starv 17448  df-sca 17449  df-vsca 17450  df-ip 17451  df-tset 17452  df-ple 17453  df-ds 17455  df-unif 17456  df-hom 17457  df-cco 17458  df-rest 17598  df-topn 17599  df-0g 17617  df-gsum 17618  df-topgen 17619  df-pt 17620  df-prds 17623  df-xrs 17679  df-qtop 17684  df-imas 17685  df-xps 17687  df-mre 17761  df-mrc 17762  df-acs 17764  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-submnd 18984  df-mulg 19283  df-cntz 19536  df-cmn 20001  df-psmet 21675  df-xmet 21676  df-met 21677  df-bl 21678  df-mopn 21679  df-cnfld 21684  df-top 23217  df-topon 23234  df-topsp 23256  df-bases 23269  df-cld 23342  df-ntr 23343  df-cls 23344  df-nei 23421  df-cn 23550  df-cnp 23551  df-cmp 23710  df-conn 23735  df-lly 23790  df-nlly 23791  df-tx 23886  df-hmeo 24079  df-xms 24644  df-ms 24645  df-tms 24646  df-ii 25203  df-cncf 25204  df-htpy 25296  df-phtpy 25297  df-phtpc 25318  df-pco 25331  df-pconn 36072  df-sconn 36073  df-cvm 36107

This theorem is used by:  cvmlift3lem6  36175  cvmlift3lem7  36176  cvmlift3lem9  36178