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Theorem fvmptrabfv 7018
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 6881 . . . 4 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2 fvmptrabfv.r . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
31, 2rabeqbidv 3450 . . 3 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
4 fvmptrabfv.f . . 3 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺𝑥) ∣ 𝜑})
5 fvex 6894 . . . 4 (𝐺𝑋) ∈ V
65rabex 5328 . . 3 {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} ∈ V
73, 4, 6fvmpt 6987 . 2 (𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
8 fvprc 6873 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
9 fvprc 6873 . . . . 5 𝑋 ∈ V → (𝐺𝑋) = ∅)
109rabeqdv 3448 . . . 4 𝑋 ∈ V → {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11 rab0 4380 . . . 4 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1210, 11eqtr2di 2790 . . 3 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
138, 12eqtrd 2773 . 2 𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
147, 13pm2.61i 182 1 (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  c0 4320  cmpt 5227  cfv 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543
This theorem is referenced by:  uvtxval  28611
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