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Theorem fvmptrab 46735
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 7032, 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 3437 . . . 4 (𝑥 = 𝑋 → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
65adantl 480 . . 3 ((𝑋𝑉𝑥 = 𝑋) → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
7 id 22 . . 3 (𝑋𝑉𝑋𝑉)
8 eqid 2725 . . . 4 {𝑦𝑁𝜓} = {𝑦𝑁𝜓}
9 fvmptrab.v . . . 4 (𝑋𝑉𝑁 ∈ V)
108, 9rabexd 5330 . . 3 (𝑋𝑉 → {𝑦𝑁𝜓} ∈ V)
112, 6, 7, 10fvmptd 7007 . 2 (𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
121fvmptndm 7031 . . 3 𝑋𝑉 → (𝐹𝑋) = ∅)
13 df-nel 3037 . . . 4 (𝑋𝑉 ↔ ¬ 𝑋𝑉)
14 fvmptrab.n . . . . 5 (𝑋𝑉𝑁 = ∅)
15 rabeq 3434 . . . . . 6 (𝑁 = ∅ → {𝑦𝑁𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16 rab0 4378 . . . . . 6 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1715, 16eqtr2di 2782 . . . . 5 (𝑁 = ∅ → ∅ = {𝑦𝑁𝜓})
1814, 17syl 17 . . . 4 (𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
1913, 18sylbir 234 . . 3 𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
2012, 19eqtrd 2765 . 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 3036  {crab 3419  Vcvv 3463  c0 4318  cmpt 5226  cfv 6543
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by: (None)
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