Theorem fsovcnvlem 40714

Description: The 𝑂 operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)

Hypotheses

Ref	Expression
fsovd.fs	⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
fsovd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsovd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fsovfvd.g	⊢ 𝐺 = (𝐴𝑂𝐵)
fsovcnvlem.h	⊢ 𝐻 = (𝐵𝑂𝐴)

Assertion

Ref	Expression
fsovcnvlem	⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑m 𝐴)))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑦   𝜑,𝑎,𝑏,𝑓,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐺(𝑥,𝑦,𝑓,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑓,𝑎,𝑏)

Proof of Theorem fsovcnvlem

Dummy variables 𝑐 𝑑 𝑔 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsovd.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		ssrab2 4007	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐴
3	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐴)
4	1, 3	sselpwd 5194	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ 𝒫 𝐴)
5	4	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ 𝒫 𝐴)
6	5	fmpttd 6856	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}):𝐵⟶𝒫 𝐴)
7	1	pwexd 5245	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
8		fsovd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9	7, 8	elmapd 8403	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}):𝐵⟶𝒫 𝐴))
10	6, 9	mpbird 260	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵))
11	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵))
12		fsovfvd.g	. . . 4 ⊢ 𝐺 = (𝐴𝑂𝐵)
13		fsovd.fs	. . . . 5 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
14	13, 1, 8	fsovd 40709	. . . 4 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
15	12, 14	syl5eq 2845	. . 3 ⊢ (𝜑 → 𝐺 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
16		fsovcnvlem.h	. . . 4 ⊢ 𝐻 = (𝐵𝑂𝐴)
17		oveq2 7143	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (𝒫 𝑏 ↑m 𝑎) = (𝒫 𝑏 ↑m 𝑑))
18		rabeq 3431	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})
19	18	mpteq2dv 5126	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
20	17, 19	mpteq12dv 5115	. . . . . . 7 ⊢ (𝑎 = 𝑑 → (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑓 ∈ (𝒫 𝑏 ↑m 𝑑) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
21		pweq 4513	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → 𝒫 𝑏 = 𝒫 𝑐)
22	21	oveq1d 7150	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝒫 𝑏 ↑m 𝑑) = (𝒫 𝑐 ↑m 𝑑))
23		mpteq1 5118	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
24	22, 23	mpteq12dv 5115	. . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝑓 ∈ (𝒫 𝑏 ↑m 𝑑) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
25	20, 24	cbvmpov 7228	. . . . . 6 ⊢ (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
26		eqid 2798	. . . . . . 7 ⊢ V = V
27		fveq1 6644	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
28	27	eleq2d 2875	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ (𝑔‘𝑥)))
29	28	rabbidv 3427	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)})
30	29	mpteq2dv 5126	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}))
31	30	cbvmptv 5133	. . . . . . . 8 ⊢ (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}))
32		eleq1w 2872	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ (𝑔‘𝑥) ↔ 𝑢 ∈ (𝑔‘𝑥)))
33	32	rabbidv 3427	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)} = {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)})
34	33	cbvmptv 5133	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}) = (𝑢 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)})
35		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (𝑔‘𝑥) = (𝑔‘𝑣))
36	35	eleq2d 2875	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑣 → (𝑢 ∈ (𝑔‘𝑥) ↔ 𝑢 ∈ (𝑔‘𝑣)))
37	36	cbvrabv 3439	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)} = {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)}
38	37	mpteq2i 5122	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)}) = (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})
39	34, 38	eqtri 2821	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}) = (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})
40	39	mpteq2i 5122	. . . . . . . 8 ⊢ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)})) = (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)}))
41	31, 40	eqtri 2821	. . . . . . 7 ⊢ (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)}))
42	26, 26, 41	mpoeq123i 7209	. . . . . 6 ⊢ (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
43	13, 25, 42	3eqtri 2825	. . . . 5 ⊢ 𝑂 = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
44	43, 8, 1	fsovd 40709	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴) = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐵) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
45	16, 44	syl5eq 2845	. . 3 ⊢ (𝜑 → 𝐻 = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐵) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
46		fveq1 6644	. . . . . 6 ⊢ (𝑔 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) → (𝑔‘𝑣) = ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣))
47	46	eleq2d 2875	. . . . 5 ⊢ (𝑔 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) → (𝑢 ∈ (𝑔‘𝑣) ↔ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)))
48	47	rabbidv 3427	. . . 4 ⊢ (𝑔 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) → {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ (𝑔‘𝑣)} = {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)})
49	48	mpteq2dv 5126	. . 3 ⊢ (𝑔 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) → (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ (𝑔‘𝑣)}) = (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)}))
50	11, 15, 45, 49	fmptco 6868	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐺) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)})))
51		eqidd 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
52		eleq1w 2872	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑣 ∈ (𝑓‘𝑥)))
53	52	rabbidv 3427	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)})
54	53	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) ∧ 𝑦 = 𝑣) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)})
55		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
56		rabexg 5198	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ∈ V)
57	1, 56	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ∈ V)
58	57	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ∈ V)
59	51, 54, 55, 58	fvmptd 6752	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣) = {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)})
60	59	eleq2d 2875	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣) ↔ 𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)}))
61		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑓‘𝑥) = (𝑓‘𝑢))
62	61	eleq2d 2875	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑣 ∈ (𝑓‘𝑥) ↔ 𝑣 ∈ (𝑓‘𝑢)))
63	62	elrab3 3629	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐴 → (𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ↔ 𝑣 ∈ (𝑓‘𝑢)))
64	63	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ↔ 𝑣 ∈ (𝑓‘𝑢)))
65	60, 64	bitrd 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣) ↔ 𝑣 ∈ (𝑓‘𝑢)))
66	65	rabbidva 3425	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)} = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)})
67	66	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)} = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)})
68		dfin5 3889	. . . . . . 7 ⊢ (𝐵 ∩ (𝑓‘𝑢)) = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)}
69		elmapi 8411	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
70	69	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
71		simpr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
72	70, 71	ffvelrnd 6829	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ∈ 𝒫 𝐵)
73	72	elpwid 4508	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ⊆ 𝐵)
74		sseqin2 4142	. . . . . . . 8 ⊢ ((𝑓‘𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢))
75	73, 74	sylib 221	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢))
76	68, 75	syl5reqr 2848	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)})
77	67, 76	eqtr4d 2836	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)} = (𝑓‘𝑢))
78	77	mpteq2dva 5125	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)}) = (𝑢 ∈ 𝐴 ↦ (𝑓‘𝑢)))
79	69	feqmptd 6708	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓 = (𝑢 ∈ 𝐴 ↦ (𝑓‘𝑢)))
80	79	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → 𝑓 = (𝑢 ∈ 𝐴 ↦ (𝑓‘𝑢)))
81	78, 80	eqtr4d 2836	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)}) = 𝑓)
82	81	mpteq2dva 5125	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ 𝑓))
83		mptresid 5885	. . . 4 ⊢ ( I ↾ (𝒫 𝐵 ↑m 𝐴)) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ 𝑓)
84	83	eqcomi 2807	. . 3 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ 𝑓) = ( I ↾ (𝒫 𝐵 ↑m 𝐴))
85	84	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ 𝑓) = ( I ↾ (𝒫 𝐵 ↑m 𝐴)))
86	50, 82, 85	3eqtrd 2837	1 ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑m 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497   ↦ cmpt 5110   I cid 5424   ↾ cres 5521   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391

This theorem is referenced by:  fsovcnvd  40715