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

Theorem fvmptrab 42195
 Description: Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 6557, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.)
Hypotheses
Ref Expression
fvmptrab.f 𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})
fvmptrab.r (𝑥 = 𝑋 → (𝜑𝜓))
fvmptrab.s (𝑥 = 𝑋𝑀 = 𝑁)
fvmptrab.v (𝑋𝑉𝑁 ∈ V)
fvmptrab.n (𝑋𝑉𝑁 = ∅)
Assertion
Ref Expression
fvmptrab (𝐹𝑋) = {𝑦𝑁𝜓}
Distinct variable groups:   𝑦,𝑀   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦   𝑥,𝑉   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐹(𝑥,𝑦)   𝑀(𝑥)   𝑉(𝑦)

Proof of Theorem fvmptrab
StepHypRef Expression
1 fvmptrab.f . . . 4 𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})
21a1i 11 . . 3 (𝑋𝑉𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑}))
3 fvmptrab.s . . . . 5 (𝑥 = 𝑋𝑀 = 𝑁)
4 fvmptrab.r . . . . 5 (𝑥 = 𝑋 → (𝜑𝜓))
53, 4rabeqbidv 3408 . . . 4 (𝑥 = 𝑋 → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
65adantl 475 . . 3 ((𝑋𝑉𝑥 = 𝑋) → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
7 id 22 . . 3 (𝑋𝑉𝑋𝑉)
8 eqid 2825 . . . 4 {𝑦𝑁𝜓} = {𝑦𝑁𝜓}
9 fvmptrab.v . . . 4 (𝑋𝑉𝑁 ∈ V)
108, 9rabexd 5038 . . 3 (𝑋𝑉 → {𝑦𝑁𝜓} ∈ V)
112, 6, 7, 10fvmptd 6535 . 2 (𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
121fvmptndm 6556 . . 3 𝑋𝑉 → (𝐹𝑋) = ∅)
13 df-nel 3103 . . . 4 (𝑋𝑉 ↔ ¬ 𝑋𝑉)
14 fvmptrab.n . . . . 5 (𝑋𝑉𝑁 = ∅)
15 rabeq 3405 . . . . . 6 (𝑁 = ∅ → {𝑦𝑁𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16 rab0 4185 . . . . . 6 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1715, 16syl6req 2878 . . . . 5 (𝑁 = ∅ → ∅ = {𝑦𝑁𝜓})
1814, 17syl 17 . . . 4 (𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
1913, 18sylbir 227 . . 3 𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
2012, 19eqtrd 2861 . 2 𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
2111, 20pm2.61i 177 1 (𝐹𝑋) = {𝑦𝑁𝜓}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1658   ∈ wcel 2166   ∉ wnel 3102  {crab 3121  Vcvv 3414  ∅c0 4144   ↦ cmpt 4952  ‘cfv 6123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131 This theorem is referenced by: (None)
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