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Theorem fsovrfovd 37782
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 6913 . . . . . 6 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
41, 2, 3syl2anc 692 . . . . 5 (𝜑 → (𝐵 × 𝐴) ∈ V)
54adantr 481 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝐵 × 𝐴) ∈ V)
6 elmapi 7823 . . . . . . . . . . . . . . 15 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
76ffvelrnda 6315 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑢𝐴) → (𝑓𝑢) ∈ 𝒫 𝐵)
87elpwid 4141 . . . . . . . . . . . . 13 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑢𝐴) → (𝑓𝑢) ⊆ 𝐵)
98sseld 3582 . . . . . . . . . . . 12 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑢𝐴) → (𝑣 ∈ (𝑓𝑢) → 𝑣𝐵))
109impancom 456 . . . . . . . . . . 11 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑣 ∈ (𝑓𝑢)) → (𝑢𝐴𝑣𝐵))
1110pm4.71d 665 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑣 ∈ (𝑓𝑢)) → (𝑢𝐴 ↔ (𝑢𝐴𝑣𝐵)))
1211ex 450 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → (𝑣 ∈ (𝑓𝑢) → (𝑢𝐴 ↔ (𝑢𝐴𝑣𝐵))))
1312pm5.32rd 671 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ ((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢))))
14 ancom 466 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) ↔ (𝑣𝐵𝑢𝐴))
1514anbi1i 730 . . . . . . . 8 (((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢)) ↔ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢)))
1613, 15syl6bb 276 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢))))
1716opabbidv 4678 . . . . . 6 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} = {⟨𝑣, 𝑢⟩ ∣ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢))})
18 opabssxp 5154 . . . . . 6 {⟨𝑣, 𝑢⟩ ∣ ((𝑣𝐵𝑢𝐴) ∧ 𝑣 ∈ (𝑓𝑢))} ⊆ (𝐵 × 𝐴)
1917, 18syl6eqss 3634 . . . . 5 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐵 × 𝐴))
2019adantl 482 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐵 × 𝐴))
215, 20sselpwd 4767 . . 3 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ∈ 𝒫 (𝐵 × 𝐴))
22 eqidd 2622 . . 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 37774 . . . 4 (𝜑 → (𝐵𝑅𝐴) = (𝑟 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣})))
25 breq 4615 . . . . . . . 8 (𝑟 = 𝑡 → (𝑢𝑟𝑣𝑢𝑡𝑣))
2625rabbidv 3177 . . . . . . 7 (𝑟 = 𝑡 → {𝑣𝐴𝑢𝑟𝑣} = {𝑣𝐴𝑢𝑡𝑣})
2726mpteq2dv 4705 . . . . . 6 (𝑟 = 𝑡 → (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣}) = (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑡𝑣}))
28 breq1 4616 . . . . . . . . 9 (𝑢 = 𝑐 → (𝑢𝑡𝑣𝑐𝑡𝑣))
2928rabbidv 3177 . . . . . . . 8 (𝑢 = 𝑐 → {𝑣𝐴𝑢𝑡𝑣} = {𝑣𝐴𝑐𝑡𝑣})
30 breq2 4617 . . . . . . . . 9 (𝑣 = 𝑑 → (𝑐𝑡𝑣𝑐𝑡𝑑))
3130cbvrabv 3185 . . . . . . . 8 {𝑣𝐴𝑐𝑡𝑣} = {𝑑𝐴𝑐𝑡𝑑}
3229, 31syl6eq 2671 . . . . . . 7 (𝑢 = 𝑐 → {𝑣𝐴𝑢𝑡𝑣} = {𝑑𝐴𝑐𝑡𝑑})
3332cbvmptv 4710 . . . . . 6 (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑡𝑣}) = (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑})
3427, 33syl6eq 2671 . . . . 5 (𝑟 = 𝑡 → (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣}) = (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}))
3534cbvmptv 4710 . . . 4 (𝑟 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑢𝐵 ↦ {𝑣𝐴𝑢𝑟𝑣})) = (𝑡 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}))
3624, 35syl6eq 2671 . . 3 (𝜑 → (𝐵𝑅𝐴) = (𝑡 ∈ 𝒫 (𝐵 × 𝐴) ↦ (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑})))
37 breq 4615 . . . . . . 7 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝑡𝑑𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}𝑑))
38 df-br 4614 . . . . . . . 8 (𝑐{⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}𝑑 ↔ ⟨𝑐, 𝑑⟩ ∈ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})
39 vex 3189 . . . . . . . . 9 𝑐 ∈ V
40 vex 3189 . . . . . . . . 9 𝑑 ∈ V
41 eleq1 2686 . . . . . . . . . 10 (𝑣 = 𝑐 → (𝑣 ∈ (𝑓𝑢) ↔ 𝑐 ∈ (𝑓𝑢)))
4241anbi2d 739 . . . . . . . . 9 (𝑣 = 𝑐 → ((𝑢𝐴𝑣 ∈ (𝑓𝑢)) ↔ (𝑢𝐴𝑐 ∈ (𝑓𝑢))))
43 eleq1 2686 . . . . . . . . . 10 (𝑢 = 𝑑 → (𝑢𝐴𝑑𝐴))
44 fveq2 6148 . . . . . . . . . . 11 (𝑢 = 𝑑 → (𝑓𝑢) = (𝑓𝑑))
4544eleq2d 2684 . . . . . . . . . 10 (𝑢 = 𝑑 → (𝑐 ∈ (𝑓𝑢) ↔ 𝑐 ∈ (𝑓𝑑)))
4643, 45anbi12d 746 . . . . . . . . 9 (𝑢 = 𝑑 → ((𝑢𝐴𝑐 ∈ (𝑓𝑢)) ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑))))
4739, 40, 42, 46opelopab 4957 . . . . . . . 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 3177 . . . . 5 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → {𝑑𝐴𝑐𝑡𝑑} = {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))})
5150mpteq2dv 4705 . . . 4 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}) = (𝑐𝐵 ↦ {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))}))
52 ibar 525 . . . . . . . . 9 (𝑑𝐴 → (𝑐 ∈ (𝑓𝑑) ↔ (𝑑𝐴𝑐 ∈ (𝑓𝑑))))
5352bicomd 213 . . . . . . . 8 (𝑑𝐴 → ((𝑑𝐴𝑐 ∈ (𝑓𝑑)) ↔ 𝑐 ∈ (𝑓𝑑)))
5453rabbiia 3173 . . . . . . 7 {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))} = {𝑑𝐴𝑐 ∈ (𝑓𝑑)}
55 fveq2 6148 . . . . . . . . 9 (𝑑 = 𝑥 → (𝑓𝑑) = (𝑓𝑥))
5655eleq2d 2684 . . . . . . . 8 (𝑑 = 𝑥 → (𝑐 ∈ (𝑓𝑑) ↔ 𝑐 ∈ (𝑓𝑥)))
5756cbvrabv 3185 . . . . . . 7 {𝑑𝐴𝑐 ∈ (𝑓𝑑)} = {𝑥𝐴𝑐 ∈ (𝑓𝑥)}
5854, 57eqtri 2643 . . . . . 6 {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))} = {𝑥𝐴𝑐 ∈ (𝑓𝑥)}
5958mpteq2i 4701 . . . . 5 (𝑐𝐵 ↦ {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))}) = (𝑐𝐵 ↦ {𝑥𝐴𝑐 ∈ (𝑓𝑥)})
60 eleq1 2686 . . . . . . 7 (𝑐 = 𝑦 → (𝑐 ∈ (𝑓𝑥) ↔ 𝑦 ∈ (𝑓𝑥)))
6160rabbidv 3177 . . . . . 6 (𝑐 = 𝑦 → {𝑥𝐴𝑐 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
6261cbvmptv 4710 . . . . 5 (𝑐𝐵 ↦ {𝑥𝐴𝑐 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
6359, 62eqtri 2643 . . . 4 (𝑐𝐵 ↦ {𝑑𝐴 ∣ (𝑑𝐴𝑐 ∈ (𝑓𝑑))}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})
6451, 63syl6eq 2671 . . 3 (𝑡 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → (𝑐𝐵 ↦ {𝑑𝐴𝑐𝑡𝑑}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}))
6521, 22, 36, 64fmptco 6351 . 2 (𝜑 → ((𝐵𝑅𝐴) ∘ (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
66 xpexg 6913 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
672, 1, 66syl2anc 692 . . . . . 6 (𝜑 → (𝐴 × 𝐵) ∈ V)
6867adantr 481 . . . . 5 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → (𝐴 × 𝐵) ∈ V)
6913opabbidv 4678 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢))})
70 opabssxp 5154 . . . . . . 7 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝐴𝑣𝐵) ∧ 𝑣 ∈ (𝑓𝑢))} ⊆ (𝐴 × 𝐵)
7169, 70syl6eqss 3634 . . . . . 6 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐴 × 𝐵))
7271adantl 482 . . . . 5 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ⊆ (𝐴 × 𝐵))
7368, 72sselpwd 4767 . . . 4 ((𝜑𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} ∈ 𝒫 (𝐴 × 𝐵))
74 eqid 2621 . . . . . 6 (𝐴𝑅𝐵) = (𝐴𝑅𝐵)
7523, 2, 1, 74rfovcnvd 37778 . . . . 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 5094 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎 × 𝑏) = (𝐴 × 𝐵))
8079pweqd 4135 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → 𝒫 (𝑎 × 𝑏) = 𝒫 (𝐴 × 𝐵))
8180mpteq1d 4698 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠))
8281adantl 482 . . . . 5 ((𝜑 ∧ (𝑎 = 𝐴𝑏 = 𝐵)) → (𝑠 ∈ 𝒫 (𝑎 × 𝑏) ↦ 𝑠) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠))
832elexd 3200 . . . . 5 (𝜑𝐴 ∈ V)
841elexd 3200 . . . . 5 (𝜑𝐵 ∈ V)
85 pwexg 4810 . . . . . 6 ((𝐴 × 𝐵) ∈ V → 𝒫 (𝐴 × 𝐵) ∈ V)
86 mptexg 6438 . . . . . 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 6741 . . . 4 (𝜑 → (𝐴𝐶𝐵) = (𝑠 ∈ 𝒫 (𝐴 × 𝐵) ↦ 𝑠))
89 cnveq 5256 . . . . 5 (𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → 𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})
90 cnvopab 5492 . . . . 5 {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}
9189, 90syl6eq 2671 . . . 4 (𝑠 = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))} → 𝑠 = {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))})
9273, 76, 88, 91fmptco 6351 . . 3 (𝜑 → ((𝐴𝐶𝐵) ∘ (𝐴𝑅𝐵)) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑣, 𝑢⟩ ∣ (𝑢𝐴𝑣 ∈ (𝑓𝑢))}))
9392coeq2d 5244 . 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 37781 . 2 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
9665, 93, 953eqtr4rd 2666 1 (𝜑 → (𝐴𝑂𝐵) = ((𝐵𝑅𝐴) ∘ ((𝐴𝐶𝐵) ∘ (𝐴𝑅𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  cop 4154   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  ccnv 5073  ccom 5078  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by: (None)
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