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Theorem fvmptrab 43498
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 6801, 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 3487 . . . 4 (𝑥 = 𝑋 → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
65adantl 484 . . 3 ((𝑋𝑉𝑥 = 𝑋) → {𝑦𝑀𝜑} = {𝑦𝑁𝜓})
7 id 22 . . 3 (𝑋𝑉𝑋𝑉)
8 eqid 2823 . . . 4 {𝑦𝑁𝜓} = {𝑦𝑁𝜓}
9 fvmptrab.v . . . 4 (𝑋𝑉𝑁 ∈ V)
108, 9rabexd 5238 . . 3 (𝑋𝑉 → {𝑦𝑁𝜓} ∈ V)
112, 6, 7, 10fvmptd 6777 . 2 (𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
121fvmptndm 6800 . . 3 𝑋𝑉 → (𝐹𝑋) = ∅)
13 df-nel 3126 . . . 4 (𝑋𝑉 ↔ ¬ 𝑋𝑉)
14 fvmptrab.n . . . . 5 (𝑋𝑉𝑁 = ∅)
15 rabeq 3485 . . . . . 6 (𝑁 = ∅ → {𝑦𝑁𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16 rab0 4339 . . . . . 6 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
1715, 16syl6req 2875 . . . . 5 (𝑁 = ∅ → ∅ = {𝑦𝑁𝜓})
1814, 17syl 17 . . . 4 (𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
1913, 18sylbir 237 . . 3 𝑋𝑉 → ∅ = {𝑦𝑁𝜓})
2012, 19eqtrd 2858 . 2 𝑋𝑉 → (𝐹𝑋) = {𝑦𝑁𝜓})
2111, 20pm2.61i 184 1 (𝐹𝑋) = {𝑦𝑁𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208   = wceq 1537  wcel 2114  wnel 3125  {crab 3144  Vcvv 3496  c0 4293  cmpt 5148  cfv 6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365
This theorem is referenced by: (None)
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