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Theorem fvmptrabfv 6780
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 6649 . . . 4 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
2 fvmptrabfv.r . . . 4 (𝑥 = 𝑋 → (𝜑𝜓))
31, 2rabeqbidv 3436 . . 3 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
4 fvmptrabfv.f . . 3 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺𝑥) ∣ 𝜑})
5 fvex 6662 . . . 4 (𝐺𝑋) ∈ V
65rabex 5202 . . 3 {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} ∈ V
73, 4, 6fvmpt 6749 . 2 (𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
8 fvprc 6642 . . 3 𝑋 ∈ V → (𝐹𝑋) = ∅)
9 fvprc 6642 . . . . 5 𝑋 ∈ V → (𝐺𝑋) = ∅)
109rabeqdv 3435 . . . 4 𝑋 ∈ V → {𝑦 ∈ (𝐺𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11 rab0 4294 . . . 4 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1210, 11eqtr2di 2853 . . 3 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
138, 12eqtrd 2836 . 2 𝑋 ∈ V → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓})
147, 13pm2.61i 185 1 (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  c0 4246  cmpt 5113  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336
This theorem is referenced by:  uvtxval  27180
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