Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvmptrabfv Structured version   Visualization version   GIF version

Theorem fvmptrabfv 6463
 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 6344 . . . 4 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2 fvmptrabfv.r . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
31, 2rabeqbidv 3327 . . 3 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
4 fvmptrabfv.f . . 3 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺𝑥) ∣ 𝜑})
5 fvex 6354 . . . 4 (𝐺𝑋) ∈ V
65rabex 4956 . . 3 {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} ∈ V
73, 4, 6fvmpt 6436 . 2 (𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
8 fvprc 6338 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
9 fvprc 6338 . . . . 5 𝑋 ∈ V → (𝐺𝑋) = ∅)
109rabeqdv 3326 . . . 4 𝑋 ∈ V → {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11 rab0 4090 . . . 4 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1210, 11syl6req 2803 . . 3 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
138, 12eqtrd 2786 . 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 196   = wceq 1624   ∈ wcel 2131  {crab 3046  Vcvv 3332  ∅c0 4050   ↦ cmpt 4873  ‘cfv 6041 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049 This theorem is referenced by:  uvtxval  26479
 Copyright terms: Public domain W3C validator