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Theorem fsovcnvfvd 41593
Description: The value of the converse of (𝐴𝑂𝐵), where 𝑂 is the 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, evaluated at function 𝐹. (Contributed by RP, 27-Apr-2021.)
Hypotheses
Ref Expression
fsovd.fs 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
fsovd.a (𝜑𝐴𝑉)
fsovd.b (𝜑𝐵𝑊)
fsovfvd.g 𝐺 = (𝐴𝑂𝐵)
fsovcnvfvd.f (𝜑𝐹 ∈ (𝒫 𝐴m 𝐵))
Assertion
Ref Expression
fsovcnvfvd (𝜑 → (𝐺𝐹) = (𝑦𝐴 ↦ {𝑥𝐵𝑦 ∈ (𝐹𝑥)}))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑥,𝑦   𝑓,𝐹,𝑥,𝑦   𝜑,𝑎,𝑏,𝑓,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑎,𝑏)   𝐺(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 fsovcnvfvd
StepHypRef Expression
1 fsovd.fs . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
2 fsovd.a . . . 4 (𝜑𝐴𝑉)
3 fsovd.b . . . 4 (𝜑𝐵𝑊)
4 fsovfvd.g . . . 4 𝐺 = (𝐴𝑂𝐵)
5 eqid 2740 . . . 4 (𝐵𝑂𝐴) = (𝐵𝑂𝐴)
61, 2, 3, 4, 5fsovcnvd 41592 . . 3 (𝜑𝐺 = (𝐵𝑂𝐴))
76fveq1d 6773 . 2 (𝜑 → (𝐺𝐹) = ((𝐵𝑂𝐴)‘𝐹))
8 fsovcnvfvd.f . . 3 (𝜑𝐹 ∈ (𝒫 𝐴m 𝐵))
91, 3, 2, 5, 8fsovfvd 41588 . 2 (𝜑 → ((𝐵𝑂𝐴)‘𝐹) = (𝑦𝐴 ↦ {𝑥𝐵𝑦 ∈ (𝐹𝑥)}))
107, 9eqtrd 2780 1 (𝜑 → (𝐺𝐹) = (𝑦𝐴 ↦ {𝑥𝐵𝑦 ∈ (𝐹𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  {crab 3070  Vcvv 3431  𝒫 cpw 4539  cmpt 5162  ccnv 5589  cfv 6432  (class class class)co 7271  cmpo 7273  m cmap 8598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-map 8600
This theorem is referenced by: (None)
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