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Theorem fvmptrab 46572
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 7023, 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 3443 . . . 4 (𝑥 = 𝑋 → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
65adantl 481 . . 3 ((𝑋𝑉𝑥 = 𝑋) → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
7 id 22 . . 3 (𝑋𝑉𝑋𝑉)
8 eqid 2726 . . . 4 {𝑦𝑁𝜓} = {𝑦𝑁𝜓}
9 fvmptrab.v . . . 4 (𝑋𝑉𝑁 ∈ V)
108, 9rabexd 5326 . . 3 (𝑋𝑉 → {𝑦𝑁𝜓} ∈ V)
112, 6, 7, 10fvmptd 6999 . 2 (𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
121fvmptndm 7022 . . 3 𝑋𝑉 → (𝐹𝑋) = ∅)
13 df-nel 3041 . . . 4 (𝑋𝑉 ↔ ¬ 𝑋𝑉)
14 fvmptrab.n . . . . 5 (𝑋𝑉𝑁 = ∅)
15 rabeq 3440 . . . . . 6 (𝑁 = ∅ → {𝑦𝑁𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16 rab0 4377 . . . . . 6 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1715, 16eqtr2di 2783 . . . . 5 (𝑁 = ∅ → ∅ = {𝑦𝑁𝜓})
1814, 17syl 17 . . . 4 (𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
1913, 18sylbir 234 . . 3 𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
2012, 19eqtrd 2766 . 2 𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
2111, 20pm2.61i 182 1 (𝐹𝑋) = {𝑦𝑁𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1533  wcel 2098  wnel 3040  {crab 3426  Vcvv 3468  c0 4317  cmpt 5224  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by: (None)
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