Theorem fsovrfovd 38821

Description: The operator which gives a 1-to-1 a mapping to a subset and a reverse mapping from elements can be composed from the operator which gives a 1-to-1 mapping between relations and functions to subsets and the converse operator. (Contributed by RP, 15-May-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)
fsovd.rf	⊢ 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢 ∈ 𝑎 ↦ {𝑣 ∈ 𝑏 ∣ 𝑢𝑟𝑣})))
fsovd.cnv	⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ◡𝑠))

Assertion

Ref	Expression
fsovrfovd	⊢ (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ ◡(𝐴𝑅𝐵))))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑟,𝑢,𝑣   𝐴,𝑠,𝑎,𝑏,𝑓,𝑢,𝑣   𝑥,𝐴,𝑦,𝑎,𝑏,𝑓   𝐵,𝑎,𝑏,𝑓,𝑟,𝑢,𝑣   𝐵,𝑠   𝑦,𝐵   𝑊,𝑎,𝑢   𝜑,𝑎,𝑏,𝑓,𝑟,𝑢,𝑣

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑠)   𝐵(𝑥)   𝐶(𝑥,𝑦,𝑣,𝑢,𝑓,𝑠,𝑟,𝑎,𝑏)   𝑅(𝑥,𝑦,𝑣,𝑢,𝑓,𝑠,𝑟,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑣,𝑢,𝑓,𝑠,𝑟,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑣,𝑢,𝑓,𝑠,𝑟,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑣,𝑓,𝑠,𝑟,𝑏)

Proof of Theorem fsovrfovd

Dummy variables 𝑐 𝑑 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsovd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
2		fsovd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		xpexg 7200	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
4	1, 2, 3	syl2anc 575	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐴) ∈ V)
5	4	adantr 468	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → (𝐵 × 𝐴) ∈ V)
6		elmapi 8124	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
7	6	ffvelrnda 6591	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ∈ 𝒫 𝐵)
8	7	elpwid 4374	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ⊆ 𝐵)
9	8	sseld 3808	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑢 ∈ 𝐴) → (𝑣 ∈ (𝑓‘𝑢) → 𝑣 ∈ 𝐵))
10	9	impancom 441	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑣 ∈ (𝑓‘𝑢)) → (𝑢 ∈ 𝐴 → 𝑣 ∈ 𝐵))
11	10	pm4.71d 553	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑣 ∈ (𝑓‘𝑢)) → (𝑢 ∈ 𝐴 ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)))
12	11	ex 399	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → (𝑣 ∈ (𝑓‘𝑢) → (𝑢 ∈ 𝐴 ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵))))
13	12	pm5.32rd 569	. . . . . . . 8 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑣 ∈ (𝑓‘𝑢))))
14		ancom 450	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ↔ (𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴))
15	14	anbi1i 612	. . . . . . . 8 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ ((𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ (𝑓‘𝑢)))
16	13, 15	syl6bb 278	. . . . . . 7 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ ((𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ (𝑓‘𝑢))))
17	16	opabbidv 4921	. . . . . 6 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} = {⟨𝑣, 𝑢⟩ ∣ ((𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ (𝑓‘𝑢))})
18		opabssxp 5409	. . . . . 6 ⊢ {⟨𝑣, 𝑢⟩ ∣ ((𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐴) ∧ 𝑣 ∈ (𝑓‘𝑢))} ⊆ (𝐵 × 𝐴)
19	17, 18	syl6eqss 3863	. . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ⊆ (𝐵 × 𝐴))
20	19	adantl 469	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ⊆ (𝐵 × 𝐴))
21	5, 20	sselpwd 5015	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ∈ 𝒫 (𝐵 × 𝐴))
22		eqidd 2818	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}))
23		fsovd.rf	. . . . 5 ⊢ 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢 ∈ 𝑎 ↦ {𝑣 ∈ 𝑏 ∣ 𝑢𝑟𝑣})))
24	23, 1, 2	rfovd 38813	. . . 4 ⊢ (𝜑 → (𝐵𝑅𝐴) = (𝑟 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑢 ∈ 𝐵 ↦ {𝑣 ∈ 𝐴 ∣ 𝑢𝑟𝑣})))
25		breq 4857	. . . . . . . 8 ⊢ (𝑟 = 𝑡 → (𝑢𝑟𝑣 ↔ 𝑢𝑡𝑣))
26	25	rabbidv 3390	. . . . . . 7 ⊢ (𝑟 = 𝑡 → {𝑣 ∈ 𝐴 ∣ 𝑢𝑟𝑣} = {𝑣 ∈ 𝐴 ∣ 𝑢𝑡𝑣})
27	26	mpteq2dv 4950	. . . . . 6 ⊢ (𝑟 = 𝑡 → (𝑢 ∈ 𝐵 ↦ {𝑣 ∈ 𝐴 ∣ 𝑢𝑟𝑣}) = (𝑢 ∈ 𝐵 ↦ {𝑣 ∈ 𝐴 ∣ 𝑢𝑡𝑣}))
28		breq1 4858	. . . . . . . . 9 ⊢ (𝑢 = 𝑐 → (𝑢𝑡𝑣 ↔ 𝑐𝑡𝑣))
29	28	rabbidv 3390	. . . . . . . 8 ⊢ (𝑢 = 𝑐 → {𝑣 ∈ 𝐴 ∣ 𝑢𝑡𝑣} = {𝑣 ∈ 𝐴 ∣ 𝑐𝑡𝑣})
30		breq2 4859	. . . . . . . . 9 ⊢ (𝑣 = 𝑑 → (𝑐𝑡𝑣 ↔ 𝑐𝑡𝑑))
31	30	cbvrabv 3400	. . . . . . . 8 ⊢ {𝑣 ∈ 𝐴 ∣ 𝑐𝑡𝑣} = {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑}
32	29, 31	syl6eq 2867	. . . . . . 7 ⊢ (𝑢 = 𝑐 → {𝑣 ∈ 𝐴 ∣ 𝑢𝑡𝑣} = {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑})
33	32	cbvmptv 4955	. . . . . 6 ⊢ (𝑢 ∈ 𝐵 ↦ {𝑣 ∈ 𝐴 ∣ 𝑢𝑡𝑣}) = (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑})
34	27, 33	syl6eq 2867	. . . . 5 ⊢ (𝑟 = 𝑡 → (𝑢 ∈ 𝐵 ↦ {𝑣 ∈ 𝐴 ∣ 𝑢𝑟𝑣}) = (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑}))
35	34	cbvmptv 4955	. . . 4 ⊢ (𝑟 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑢 ∈ 𝐵 ↦ {𝑣 ∈ 𝐴 ∣ 𝑢𝑟𝑣})) = (𝑡 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑}))
36	24, 35	syl6eq 2867	. . 3 ⊢ (𝜑 → (𝐵𝑅𝐴) = (𝑡 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑})))
37		breq 4857	. . . . . . 7 ⊢ (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → (𝑐𝑡𝑑 ↔ 𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}𝑑))
38		df-br 4856	. . . . . . . 8 ⊢ (𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}𝑑 ↔ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))})
39		vex 3405	. . . . . . . . 9 ⊢ 𝑐 ∈ V
40		vex 3405	. . . . . . . . 9 ⊢ 𝑑 ∈ V
41		eleq1w 2879	. . . . . . . . . 10 ⊢ (𝑣 = 𝑐 → (𝑣 ∈ (𝑓‘𝑢) ↔ 𝑐 ∈ (𝑓‘𝑢)))
42	41	anbi2d 616	. . . . . . . . 9 ⊢ (𝑣 = 𝑐 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑢))))
43		eleq1w 2879	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → (𝑢 ∈ 𝐴 ↔ 𝑑 ∈ 𝐴))
44		fveq2 6418	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑑 → (𝑓‘𝑢) = (𝑓‘𝑑))
45	44	eleq2d 2882	. . . . . . . . . 10 ⊢ (𝑢 = 𝑑 → (𝑐 ∈ (𝑓‘𝑢) ↔ 𝑐 ∈ (𝑓‘𝑑)))
46	43, 45	anbi12d 618	. . . . . . . . 9 ⊢ (𝑢 = 𝑑 → ((𝑢 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑢)) ↔ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))))
47	39, 40, 42, 46	opelopab 5205	. . . . . . . 8 ⊢ (⟨𝑐, 𝑑⟩ ∈ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ↔ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑)))
48	38, 47	bitri 266	. . . . . . 7 ⊢ (𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}𝑑 ↔ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑)))
49	37, 48	syl6bb 278	. . . . . 6 ⊢ (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → (𝑐𝑡𝑑 ↔ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))))
50	49	rabbidv 3390	. . . . 5 ⊢ (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑} = {𝑑 ∈ 𝐴 ∣ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))})
51	50	mpteq2dv 4950	. . . 4 ⊢ (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑}) = (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))}))
52		ibar 520	. . . . . . . . 9 ⊢ (𝑑 ∈ 𝐴 → (𝑐 ∈ (𝑓‘𝑑) ↔ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))))
53	52	bicomd 214	. . . . . . . 8 ⊢ (𝑑 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑)) ↔ 𝑐 ∈ (𝑓‘𝑑)))
54	53	rabbiia 3385	. . . . . . 7 ⊢ {𝑑 ∈ 𝐴 ∣ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))} = {𝑑 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑑)}
55		fveq2 6418	. . . . . . . . 9 ⊢ (𝑑 = 𝑥 → (𝑓‘𝑑) = (𝑓‘𝑥))
56	55	eleq2d 2882	. . . . . . . 8 ⊢ (𝑑 = 𝑥 → (𝑐 ∈ (𝑓‘𝑑) ↔ 𝑐 ∈ (𝑓‘𝑥)))
57	56	cbvrabv 3400	. . . . . . 7 ⊢ {𝑑 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑑)} = {𝑥 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑥)}
58	54, 57	eqtri 2839	. . . . . 6 ⊢ {𝑑 ∈ 𝐴 ∣ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))} = {𝑥 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑥)}
59	58	mpteq2i 4946	. . . . 5 ⊢ (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))}) = (𝑐 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑥)})
60		eleq1w 2879	. . . . . . 7 ⊢ (𝑐 = 𝑦 → (𝑐 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ (𝑓‘𝑥)))
61	60	rabbidv 3390	. . . . . 6 ⊢ (𝑐 = 𝑦 → {𝑥 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})
62	61	cbvmptv 4955	. . . . 5 ⊢ (𝑐 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑐 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})
63	59, 62	eqtri 2839	. . . 4 ⊢ (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ (𝑓‘𝑑))}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})
64	51, 63	syl6eq 2867	. . 3 ⊢ (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → (𝑐 ∈ 𝐵 ↦ {𝑑 ∈ 𝐴 ∣ 𝑐𝑡𝑑}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))
65	21, 22, 36, 64	fmptco 6629	. 2 ⊢ (𝜑 → ((𝐵𝑅𝐴) ∘ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))})) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
66		xpexg 7200	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
67	2, 1, 66	syl2anc 575	. . . . . 6 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)
68	67	adantr 468	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → (𝐴 × 𝐵) ∈ V)
69	13	opabbidv 4921	. . . . . . 7 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑣 ∈ (𝑓‘𝑢))})
70		opabssxp 5409	. . . . . . 7 ⊢ {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵) ∧ 𝑣 ∈ (𝑓‘𝑢))} ⊆ (𝐴 × 𝐵)
71	69, 70	syl6eqss 3863	. . . . . 6 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ⊆ (𝐴 × 𝐵))
72	71	adantl 469	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ⊆ (𝐴 × 𝐵))
73	68, 72	sselpwd 5015	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) → {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} ∈ 𝒫 (𝐴 × 𝐵))
74		eqid 2817	. . . . . 6 ⊢ (𝐴𝑅𝐵) = (𝐴𝑅𝐵)
75	23, 2, 1, 74	rfovcnvd 38817	. . . . 5 ⊢ (𝜑 → ◡(𝐴𝑅𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}))
76	75	idi 2	. . . 4 ⊢ (𝜑 → ◡(𝐴𝑅𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}))
77		fsovd.cnv	. . . . . 6 ⊢ 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ◡𝑠))
78	77	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ◡𝑠)))
79		xpeq12 5348	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
80	79	pweqd 4367	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
81	80	mpteq1d 4943	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ◡𝑠) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ ◡𝑠))
82	81	adantl 469	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ ◡𝑠) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ ◡𝑠))
83	2	elexd 3419	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
84	1	elexd 3419	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
85		pwexg 5061	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
86		mptexg 6719	. . . . . 6 ⊢ (𝒫 (𝐴 × 𝐵) ∈ V → (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ ◡𝑠) ∈ V)
87	67, 85, 86	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ ◡𝑠) ∈ V)
88	78, 82, 83, 84, 87	ovmpt2d 7028	. . . 4 ⊢ (𝜑 → (𝐴𝐶𝐵) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ ◡𝑠))
89		cnveq 5511	. . . . 5 ⊢ (𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → ◡𝑠 = ◡{⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))})
90		cnvopab 5758	. . . . 5 ⊢ ◡{⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}
91	89, 90	syl6eq 2867	. . . 4 ⊢ (𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))} → ◡𝑠 = {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))})
92	73, 76, 88, 91	fmptco 6629	. . 3 ⊢ (𝜑 → ((𝐴𝐶𝐵) ∘ ◡(𝐴𝑅𝐵)) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))}))
93	92	coeq2d 5500	. 2 ⊢ (𝜑 → ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ ◡(𝐴𝑅𝐵))) = ((𝐵𝑅𝐴) ∘ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))})))
94		fsovd.fs	. . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
95	94, 2, 1	fsovd 38820	. 2 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
96	65, 93, 95	3eqtr4rd 2862	1 ⊢ (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ ◡(𝐴𝑅𝐵))))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637   ∈ wcel 2157  {crab 3111  Vcvv 3402   ⊆ wss 3780  𝒫 cpw 4362  ⟨cop 4387   class class class wbr 4855  {copab 4917   ↦ cmpt 4934   × cxp 5322  ^◡ccnv 5323   ∘ ccom 5328  ‘cfv 6111  (class class class)co 6884   ↦ cmpt2 6886   ↑_𝑚 cmap 8102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-1st 7408  df-2nd 7409  df-map 8104

This theorem is referenced by: (None)