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Theorem fvmptrabfv 6498
Description: Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.)
Hypotheses
Ref Expression
fvmptrabfv.f 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺𝑥) ∣ 𝜑})
fvmptrabfv.r (𝑥 = 𝑋 → (𝜑𝜓))
Assertion
Ref Expression
fvmptrabfv (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓}
Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fvmptrabfv
StepHypRef Expression
1 fveq2 6375 . . . 4 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2 fvmptrabfv.r . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
31, 2rabeqbidv 3344 . . 3 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
4 fvmptrabfv.f . . 3 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺𝑥) ∣ 𝜑})
5 fvex 6388 . . . 4 (𝐺𝑋) ∈ V
65rabex 4973 . . 3 {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} ∈ V
73, 4, 6fvmpt 6471 . 2 (𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
8 fvprc 6368 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
9 fvprc 6368 . . . . 5 𝑋 ∈ V → (𝐺𝑋) = ∅)
109rabeqdv 3343 . . . 4 𝑋 ∈ V → {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11 rab0 4120 . . . 4 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1210, 11syl6req 2816 . . 3 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
138, 12eqtrd 2799 . 2 𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
147, 13pm2.61i 176 1 (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  c0 4079  cmpt 4888  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076
This theorem is referenced by:  uvtxval  26568
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