Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvmptrabdm Structured version   Visualization version   GIF version

Theorem fvmptrabdm 42068
Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6502. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.)
Hypotheses
Ref Expression
fvmptrabdm.f 𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})
fvmptrabdm.r (𝑥 = 𝑋 → (𝜑𝜓))
fvmptrabdm.v (𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)
Assertion
Ref Expression
fvmptrabdm (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐹(𝑦)   𝑉(𝑦)

Proof of Theorem fvmptrabdm
StepHypRef Expression
1 fvmptrabdm.v . 2 (𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)
2 pm2.1 920 . 2 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹)
3 imor 879 . . 3 ((𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) ↔ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹))
4 ordir 1029 . . . . 5 (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) ↔ ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)))
5 ndmfv 6409 . . . . . . 7 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
6 ndmfv 6409 . . . . . . . . 9 𝑌 ∈ dom 𝐺 → (𝐺𝑌) = ∅)
76rabeqdv 3343 . . . . . . . 8 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8 rab0 4122 . . . . . . . 8 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
97, 8syl6req 2816 . . . . . . 7 𝑌 ∈ dom 𝐺 → ∅ = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
105, 9sylan9eq 2819 . . . . . 6 ((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
11 fvmptrabdm.f . . . . . . . 8 𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})
1211a1i 11 . . . . . . 7 (𝑋 ∈ dom 𝐹𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑}))
13 fvmptrabdm.r . . . . . . . . 9 (𝑥 = 𝑋 → (𝜑𝜓))
1413rabbidv 3338 . . . . . . . 8 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
1514adantl 473 . . . . . . 7 ((𝑋 ∈ dom 𝐹𝑥 = 𝑋) → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
1611dmmpt 5818 . . . . . . . . . 10 dom 𝐹 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V}
17 rabid2 3266 . . . . . . . . . . 11 (𝑉 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥𝑉 {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V)
18 fvex 6392 . . . . . . . . . . . . 13 (𝐺𝑌) ∈ V
1918rabex 4975 . . . . . . . . . . . 12 {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝑥𝑉 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V)
2117, 20mprgbir 3074 . . . . . . . . . 10 𝑉 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V}
2216, 21eqtr4i 2790 . . . . . . . . 9 dom 𝐹 = 𝑉
2322eleq2i 2836 . . . . . . . 8 (𝑋 ∈ dom 𝐹𝑋𝑉)
2423biimpi 207 . . . . . . 7 (𝑋 ∈ dom 𝐹𝑋𝑉)
2518rabex 4975 . . . . . . . 8 {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} ∈ V
2625a1i 11 . . . . . . 7 (𝑋 ∈ dom 𝐹 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} ∈ V)
2712, 15, 24, 26fvmptd 6481 . . . . . 6 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
2810, 27jaoi 883 . . . . 5 (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
294, 28sylbir 226 . . . 4 (((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
3029expcom 402 . . 3 ((¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}))
313, 30sylbi 208 . 2 ((𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}))
321, 2, 31mp2 9 1 (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  c0 4081  cmpt 4890  dom cdm 5279  cfv 6070
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 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064
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 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fv 6078
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator