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Theorem fsovfvd 39260
Description: Value of the operator, (𝐴𝑂𝐵), 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, 𝐴 and 𝐵, when applied to function 𝐹. (Contributed by RP, 25-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)
fsovfvd.f (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝐴))
Assertion
Ref Expression
fsovfvd (𝜑 → (𝐺𝐹) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥   𝑦,𝐴,𝑎,𝑏,𝑓   𝐵,𝑎,𝑏,𝑓,𝑦   𝑓,𝐹,𝑥   𝑦,𝐹   𝜑,𝑎,𝑏,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(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 fsovfvd
StepHypRef Expression
1 fsovfvd.g . . 3 𝐺 = (𝐴𝑂𝐵)
2 fsovd.fs . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
3 fsovd.a . . . 4 (𝜑𝐴𝑉)
4 fsovd.b . . . 4 (𝜑𝐵𝑊)
52, 3, 4fsovd 39258 . . 3 (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
61, 5syl5eq 2826 . 2 (𝜑𝐺 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)})))
7 fveq1 6445 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
87eleq2d 2845 . . . . 5 (𝑓 = 𝐹 → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ (𝐹𝑥)))
98rabbidv 3386 . . . 4 (𝑓 = 𝐹 → {𝑥𝐴𝑦 ∈ (𝑓𝑥)} = {𝑥𝐴𝑦 ∈ (𝐹𝑥)})
109mpteq2dv 4980 . . 3 (𝑓 = 𝐹 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
1110adantl 475 . 2 ((𝜑𝑓 = 𝐹) → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝑓𝑥)}) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
12 fsovfvd.f . 2 (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝐴))
13 mptexg 6756 . . 3 (𝐵𝑊 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}) ∈ V)
144, 13syl 17 . 2 (𝜑 → (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}) ∈ V)
156, 11, 12, 14fvmptd 6548 1 (𝜑 → (𝐺𝐹) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  {crab 3094  Vcvv 3398  𝒫 cpw 4379  cmpt 4965  cfv 6135  (class class class)co 6922  cmpt2 6924  𝑚 cmap 8140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927
This theorem is referenced by:  fsovfvfvd  39261  fsovcnvfvd  39265
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