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Theorem fsovrfovd 38620
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 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
fsovd.a (𝜑𝐴𝑉)
fsovd.b (𝜑𝐵𝑊)
fsovd.rf 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢𝑎 ↦ {𝑣𝑏𝑢𝑟𝑣})))
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.
StepHypRef Expression
1 fsovd.b . . . . . 6 (𝜑𝐵𝑊)
2 fsovd.a . . . . . 6 (𝜑𝐴𝑉)
3 xpexg 7002 . . . . . 6 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
41, 2, 3syl2anc 694 . . . . 5 (𝜑 → (𝐵 × 𝐴) ∈ V)
54adantr 480 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝐵 × 𝐴) ∈ V)
6 elmapi 7921 . . . . . . . . . . . . . . 15 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
76ffvelrnda 6399 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑢𝐴) → (𝑓𝑢) ∈ 𝒫 𝐵)
87elpwid 4203 . . . . . . . . . . . . 13 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑢𝐴) → (𝑓𝑢) ⊆ 𝐵)
98sseld 3635 . . . . . . . . . . . 12 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑢𝐴) → (𝑣 ∈ (𝑓𝑢) → 𝑣𝐵))
109impancom 455 . . . . . . . . . . 11 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑣 ∈ (𝑓𝑢)) → (𝑢𝐴𝑣𝐵))
1110pm4.71d 667 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑣 ∈ (𝑓𝑢)) → (𝑢𝐴 ↔ (𝑢𝐴𝑣𝐵)))
1211ex 449 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → (𝑣 ∈ (𝑓𝑢) → (𝑢𝐴 ↔ (𝑢𝐴𝑣𝐵))))
1312pm5.32rd 673 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ ((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢))))
14 ancom 465 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) ↔ (𝑣𝐵𝑢𝐴))
1514anbi1i 731 . . . . . . . 8 (((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢)) ↔ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢)))
1613, 15syl6bb 276 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢))))
1716opabbidv 4749 . . . . . 6 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} = {⟨𝑣, 𝑢⟩ ∣ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢))})
18 opabssxp 5227 . . . . . 6 {⟨𝑣, 𝑢⟩ ∣ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢))} ⊆ (𝐵 × 𝐴)
1917, 18syl6eqss 3688 . . . . 5 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐵 × 𝐴))
2019adantl 481 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐵 × 𝐴))
215, 20sselpwd 4840 . . 3 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ∈ 𝒫 (𝐵 × 𝐴))
22 eqidd 2652 . . 3 (𝜑 → (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}))
23 fsovd.rf . . . . 5 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑢𝑎 ↦ {𝑣𝑏𝑢𝑟𝑣})))
2423, 1, 2rfovd 38612 . . . 4 (𝜑 → (𝐵𝑅𝐴) = (𝑟 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣})))
25 breq 4687 . . . . . . . 8 (𝑟 = 𝑡 → (𝑢𝑟𝑣𝑢𝑡𝑣))
2625rabbidv 3220 . . . . . . 7 (𝑟 = 𝑡 → {𝑣𝐴𝑢𝑟𝑣} = {𝑣𝐴𝑢𝑡𝑣})
2726mpteq2dv 4778 . . . . . 6 (𝑟 = 𝑡 → (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣}) = (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑡𝑣}))
28 breq1 4688 . . . . . . . . 9 (𝑢 = 𝑐 → (𝑢𝑡𝑣𝑐𝑡𝑣))
2928rabbidv 3220 . . . . . . . 8 (𝑢 = 𝑐 → {𝑣𝐴𝑢𝑡𝑣} = {𝑣𝐴𝑐𝑡𝑣})
30 breq2 4689 . . . . . . . . 9 (𝑣 = 𝑑 → (𝑐𝑡𝑣𝑐𝑡𝑑))
3130cbvrabv 3230 . . . . . . . 8 {𝑣𝐴𝑐𝑡𝑣} = {𝑑𝐴𝑐𝑡𝑑}
3229, 31syl6eq 2701 . . . . . . 7 (𝑢 = 𝑐 → {𝑣𝐴𝑢𝑡𝑣} = {𝑑𝐴𝑐𝑡𝑑})
3332cbvmptv 4783 . . . . . 6 (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑡𝑣}) = (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑})
3427, 33syl6eq 2701 . . . . 5 (𝑟 = 𝑡 → (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣}) = (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}))
3534cbvmptv 4783 . . . 4 (𝑟 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣})) = (𝑡 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}))
3624, 35syl6eq 2701 . . 3 (𝜑 → (𝐵𝑅𝐴) = (𝑡 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑})))
37 breq 4687 . . . . . . 7 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝑡𝑑𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}𝑑))
38 df-br 4686 . . . . . . . 8 (𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}𝑑 ↔ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})
39 vex 3234 . . . . . . . . 9 𝑐 ∈ V
40 vex 3234 . . . . . . . . 9 𝑑 ∈ V
41 eleq1 2718 . . . . . . . . . 10 (𝑣 = 𝑐 → (𝑣 ∈ (𝑓𝑢) ↔ 𝑐 ∈ (𝑓𝑢)))
4241anbi2d 740 . . . . . . . . 9 (𝑣 = 𝑐 → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ (𝑢𝐴𝑐 ∈ (𝑓𝑢))))
43 eleq1 2718 . . . . . . . . . 10 (𝑢 = 𝑑 → (𝑢𝐴𝑑𝐴))
44 fveq2 6229 . . . . . . . . . . 11 (𝑢 = 𝑑 → (𝑓𝑢) = (𝑓𝑑))
4544eleq2d 2716 . . . . . . . . . 10 (𝑢 = 𝑑 → (𝑐 ∈ (𝑓𝑢) ↔ 𝑐 ∈ (𝑓𝑑)))
4643, 45anbi12d 747 . . . . . . . . 9 (𝑢 = 𝑑 → ((𝑢𝐴𝑐 ∈ (𝑓𝑢)) ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑))))
4739, 40, 42, 46opelopab 5026 . . . . . . . 8 (⟨𝑐, 𝑑⟩ ∈ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑)))
4838, 47bitri 264 . . . . . . 7 (𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}𝑑 ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑)))
4937, 48syl6bb 276 . . . . . 6 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝑡𝑑 ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑))))
5049rabbidv 3220 . . . . 5 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → {𝑑𝐴𝑐𝑡𝑑} = {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))})
5150mpteq2dv 4778 . . . 4 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}) = (𝑐𝐵 ↦ {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))}))
52 ibar 524 . . . . . . . . 9 (𝑑𝐴 → (𝑐 ∈ (𝑓𝑑) ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑))))
5352bicomd 213 . . . . . . . 8 (𝑑𝐴 → ((𝑑𝐴𝑐 ∈ (𝑓𝑑)) ↔ 𝑐 ∈ (𝑓𝑑)))
5453rabbiia 3215 . . . . . . 7 {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))} = {𝑑𝐴𝑐 ∈ (𝑓𝑑)}
55 fveq2 6229 . . . . . . . . 9 (𝑑 = 𝑥 → (𝑓𝑑) = (𝑓𝑥))
5655eleq2d 2716 . . . . . . . 8 (𝑑 = 𝑥 → (𝑐 ∈ (𝑓𝑑) ↔ 𝑐 ∈ (𝑓𝑥)))
5756cbvrabv 3230 . . . . . . 7 {𝑑𝐴𝑐 ∈ (𝑓𝑑)} = {𝑥𝐴𝑐 ∈ (𝑓𝑥)}
5854, 57eqtri 2673 . . . . . 6 {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))} = {𝑥𝐴𝑐 ∈ (𝑓𝑥)}
5958mpteq2i 4774 . . . . 5 (𝑐𝐵 ↦ {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))}) = (𝑐𝐵 ↦ {𝑥𝐴𝑐 ∈ (𝑓𝑥)})
60 eleq1 2718 . . . . . . 7 (𝑐 = 𝑦 → (𝑐 ∈ (𝑓𝑥) ↔ 𝑦 ∈ (𝑓𝑥)))
6160rabbidv 3220 . . . . . 6 (𝑐 = 𝑦 → {𝑥𝐴𝑐 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
6261cbvmptv 4783 . . . . 5 (𝑐𝐵 ↦ {𝑥𝐴𝑐 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
6359, 62eqtri 2673 . . . 4 (𝑐𝐵 ↦ {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
6451, 63syl6eq 2701 . . 3 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}))
6521, 22, 36, 64fmptco 6436 . 2 (𝜑 → ((𝐵𝑅𝐴) ∘ (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
66 xpexg 7002 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
672, 1, 66syl2anc 694 . . . . . 6 (𝜑 → (𝐴 × 𝐵) ∈ V)
6867adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝐴 × 𝐵) ∈ V)
6913opabbidv 4749 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢))})
70 opabssxp 5227 . . . . . . 7 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢))} ⊆ (𝐴 × 𝐵)
7169, 70syl6eqss 3688 . . . . . 6 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐴 × 𝐵))
7271adantl 481 . . . . 5 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐴 × 𝐵))
7368, 72sselpwd 4840 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ∈ 𝒫 (𝐴 × 𝐵))
74 eqid 2651 . . . . . 6 (𝐴𝑅𝐵) = (𝐴𝑅𝐵)
7523, 2, 1, 74rfovcnvd 38616 . . . . 5 (𝜑(𝐴𝑅𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}))
7675idi 2 . . . 4 (𝜑(𝐴𝑅𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}))
77 fsovd.cnv . . . . . 6 𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠))
7877a1i 11 . . . . 5 (𝜑𝐶 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠)))
79 xpeq12 5168 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
8079pweqd 4196 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
8180mpteq1d 4771 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠))
8281adantl 481 . . . . 5 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠))
832elexd 3245 . . . . 5 (𝜑𝐴 ∈ V)
841elexd 3245 . . . . 5 (𝜑𝐵 ∈ V)
85 pwexg 4880 . . . . . 6 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
86 mptexg 6525 . . . . . 6 (𝒫 (𝐴 × 𝐵) ∈ V → (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠) ∈ V)
8767, 85, 863syl 18 . . . . 5 (𝜑 → (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠) ∈ V)
8878, 82, 83, 84, 87ovmpt2d 6830 . . . 4 (𝜑 → (𝐴𝐶𝐵) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠))
89 cnveq 5328 . . . . 5 (𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → 𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})
90 cnvopab 5568 . . . . 5 {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}
9189, 90syl6eq 2701 . . . 4 (𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → 𝑠 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})
9273, 76, 88, 91fmptco 6436 . . 3 (𝜑 → ((𝐴𝐶𝐵) ∘ (𝐴𝑅𝐵)) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}))
9392coeq2d 5317 . 2 (𝜑 → ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ (𝐴𝑅𝐵))) = ((𝐵𝑅𝐴) ∘ (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})))
94 fsovd.fs . . 3 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
9594, 2, 1fsovd 38619 . 2 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
9665, 93, 953eqtr4rd 2696 1 (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ (𝐴𝑅𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  cop 4216   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  ccnv 5142  ccom 5147  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901
This theorem is referenced by: (None)
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