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Theorem fvmptrab 47242
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 7048, 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 3452 . . . 4 (𝑥 = 𝑋 → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
65adantl 481 . . 3 ((𝑋𝑉𝑥 = 𝑋) → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
7 id 22 . . 3 (𝑋𝑉𝑋𝑉)
8 eqid 2735 . . . 4 {𝑦𝑁𝜓} = {𝑦𝑁𝜓}
9 fvmptrab.v . . . 4 (𝑋𝑉𝑁 ∈ V)
108, 9rabexd 5346 . . 3 (𝑋𝑉 → {𝑦𝑁𝜓} ∈ V)
112, 6, 7, 10fvmptd 7023 . 2 (𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
121fvmptndm 7047 . . 3 𝑋𝑉 → (𝐹𝑋) = ∅)
13 df-nel 3045 . . . 4 (𝑋𝑉 ↔ ¬ 𝑋𝑉)
14 fvmptrab.n . . . . 5 (𝑋𝑉𝑁 = ∅)
15 rabeq 3448 . . . . . 6 (𝑁 = ∅ → {𝑦𝑁𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16 rab0 4392 . . . . . 6 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1715, 16eqtr2di 2792 . . . . 5 (𝑁 = ∅ → ∅ = {𝑦𝑁𝜓})
1814, 17syl 17 . . . 4 (𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
1913, 18sylbir 235 . . 3 𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
2012, 19eqtrd 2775 . 2 𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
2111, 20pm2.61i 182 1 (𝐹𝑋) = {𝑦𝑁𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  wnel 3044  {crab 3433  Vcvv 3478  c0 4339  cmpt 5231  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by: (None)
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