Theorem fsovcnvlem 43975

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 4039	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐴
3	2	a1i 11	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐴)
4	1, 3	sselpwd 5278	. . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ 𝒫 𝐴)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ 𝒫 𝐴)
6	5	fmpttd 7069	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}):𝐵⟶𝒫 𝐴)
7	1	pwexd 5329	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐴 ∈ V)
8		fsovd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
9	7, 8	elmapd 8790	. . . . 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 257	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵))
11	10	adantr 480	. . 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 43970	. . . 4 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
15	12, 14	eqtrid 2776	. . 3 ⊢ (𝜑 → 𝐺 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
16		fsovcnvlem.h	. . . 4 ⊢ 𝐻 = (𝐵𝑂𝐴)
17		oveq2 7377	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (𝒫 𝑏 ↑m 𝑎) = (𝒫 𝑏 ↑m 𝑑))
18		rabeq 3417	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)})
19	18	mpteq2dv 5196	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
20	17, 19	mpteq12dv 5189	. . . . . . 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 4573	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → 𝒫 𝑏 = 𝒫 𝑐)
22	21	oveq1d 7384	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝒫 𝑏 ↑m 𝑑) = (𝒫 𝑐 ↑m 𝑑))
23		mpteq1 5191	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
24	22, 23	mpteq12dv 5189	. . . . . . 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 7464	. . . . . 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 2729	. . . . . . 7 ⊢ V = V
27		fveq1 6839	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
28	27	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ (𝑔‘𝑥)))
29	28	rabbidv 3410	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)})
30	29	mpteq2dv 5196	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}))
31	30	cbvmptv 5206	. . . . . . . 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 2811	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ (𝑔‘𝑥) ↔ 𝑢 ∈ (𝑔‘𝑥)))
33	32	rabbidv 3410	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)} = {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)})
34	33	cbvmptv 5206	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}) = (𝑢 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)})
35		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑣 → (𝑔‘𝑥) = (𝑔‘𝑣))
36	35	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑣 → (𝑢 ∈ (𝑔‘𝑥) ↔ 𝑢 ∈ (𝑔‘𝑣)))
37	36	cbvrabv 3413	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)} = {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)}
38	37	mpteq2i 5198	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑥)}) = (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})
39	34, 38	eqtri 2752	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑐 ↦ {𝑥 ∈ 𝑑 ∣ 𝑦 ∈ (𝑔‘𝑥)}) = (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})
40	39	mpteq2i 5198	. . . . . . . 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 2752	. . . . . . 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 7445	. . . . . 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 2756	. . . . 5 ⊢ 𝑂 = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑢 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
44	43, 8, 1	fsovd 43970	. . . 4 ⊢ (𝜑 → (𝐵𝑂𝐴) = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐵) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
45	16, 44	eqtrid 2776	. . 3 ⊢ (𝜑 → 𝐻 = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐵) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ (𝑔‘𝑣)})))
46		fveq1 6839	. . . . . 6 ⊢ (𝑔 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) → (𝑔‘𝑣) = ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣))
47	46	eleq2d 2814	. . . . 5 ⊢ (𝑔 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) → (𝑢 ∈ (𝑔‘𝑣) ↔ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)))
48	47	rabbidv 3410	. . . 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 5196	. . 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 7083	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝐺) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)})))
51		eqidd 2730	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
52		eleq1w 2811	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑣 → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑣 ∈ (𝑓‘𝑥)))
53	52	rabbidv 3410	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)})
54	53	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) ∧ 𝑦 = 𝑣) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)})
55		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ 𝐵)
56		rabexg 5287	. . . . . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ∈ V)
59	51, 54, 55, 58	fvmptd 6957	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣) = {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)})
60	59	eleq2d 2814	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣) ↔ 𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)}))
61		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑢 → (𝑓‘𝑥) = (𝑓‘𝑢))
62	61	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝑣 ∈ (𝑓‘𝑥) ↔ 𝑣 ∈ (𝑓‘𝑢)))
63	62	elrab3 3657	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝐴 → (𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ↔ 𝑣 ∈ (𝑓‘𝑢)))
64	63	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑣 ∈ (𝑓‘𝑥)} ↔ 𝑣 ∈ (𝑓‘𝑢)))
65	60, 64	bitrd 279	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ 𝐵) → (𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣) ↔ 𝑣 ∈ (𝑓‘𝑢)))
66	65	rabbidva 3409	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴) → {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)} = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)})
67	66	adantlr 715	. . . . . 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		elmapi 8799	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
69	68	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
70		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
71	69, 70	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ∈ 𝒫 𝐵)
72	71	elpwid 4568	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ⊆ 𝐵)
73		sseqin2 4182	. . . . . . . 8 ⊢ ((𝑓‘𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢))
74	72, 73	sylib 218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢))
75		dfin5 3919	. . . . . . 7 ⊢ (𝐵 ∩ (𝑓‘𝑢)) = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)}
76	74, 75	eqtr3di 2779	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = {𝑣 ∈ 𝐵 ∣ 𝑣 ∈ (𝑓‘𝑢)})
77	67, 76	eqtr4d 2767	. . . . 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 5195	. . . 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	68	feqmptd 6911	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓 = (𝑢 ∈ 𝐴 ↦ (𝑓‘𝑢)))
80	79	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → 𝑓 = (𝑢 ∈ 𝐴 ↦ (𝑓‘𝑢)))
81	78, 80	eqtr4d 2767	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑢 ∈ 𝐴 ↦ {𝑣 ∈ 𝐵 ∣ 𝑢 ∈ ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})‘𝑣)}) = 𝑓)
82	81	mpteq2dva 5195	. 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 6011	. . . 4 ⊢ ( I ↾ (𝒫 𝐵 ↑m 𝐴)) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ 𝑓)
84	83	eqcomi 2738	. . 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 2768	1 ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑m 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559   ↦ cmpt 5183   I cid 5525   ↾ cres 5633   ∘ ccom 5635  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ∈ cmpo 7371   ↑_m cmap 8776

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778

This theorem is referenced by:  fsovcnvd  43976