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Theorem fvmptrabdm 43377
Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6797. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.)
Hypotheses
Ref Expression
fvmptrabdm.f 𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})
fvmptrabdm.r (𝑥 = 𝑋 → (𝜑𝜓))
fvmptrabdm.v (𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)
Assertion
Ref Expression
fvmptrabdm (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐹(𝑦)   𝑉(𝑦)

Proof of Theorem fvmptrabdm
StepHypRef Expression
1 fvmptrabdm.v . 2 (𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)
2 pm2.1 892 . 2 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹)
3 imor 849 . . 3 ((𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) ↔ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹))
4 ordir 1002 . . . . 5 (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) ↔ ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)))
5 ndmfv 6699 . . . . . . 7 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
6 ndmfv 6699 . . . . . . . . 9 𝑌 ∈ dom 𝐺 → (𝐺𝑌) = ∅)
76rabeqdv 3490 . . . . . . . 8 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8 rab0 4341 . . . . . . . 8 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
97, 8syl6req 2878 . . . . . . 7 𝑌 ∈ dom 𝐺 → ∅ = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
105, 9sylan9eq 2881 . . . . . 6 ((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
11 fvmptrabdm.f . . . . . . 7 𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})
12 fvmptrabdm.r . . . . . . . 8 (𝑥 = 𝑋 → (𝜑𝜓))
1312rabbidv 3486 . . . . . . 7 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
1411dmmpt 6093 . . . . . . . . . 10 dom 𝐹 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V}
15 rabid2 3387 . . . . . . . . . . 11 (𝑉 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥𝑉 {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V)
16 fvex 6682 . . . . . . . . . . . . 13 (𝐺𝑌) ∈ V
1716rabex 5232 . . . . . . . . . . . 12 {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V
1817a1i 11 . . . . . . . . . . 11 (𝑥𝑉 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V)
1915, 18mprgbir 3158 . . . . . . . . . 10 𝑉 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V}
2014, 19eqtr4i 2852 . . . . . . . . 9 dom 𝐹 = 𝑉
2120eleq2i 2909 . . . . . . . 8 (𝑋 ∈ dom 𝐹𝑋𝑉)
2221biimpi 217 . . . . . . 7 (𝑋 ∈ dom 𝐹𝑋𝑉)
2316rabex 5232 . . . . . . . 8 {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} ∈ V
2423a1i 11 . . . . . . 7 (𝑋 ∈ dom 𝐹 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} ∈ V)
2511, 13, 22, 24fvmptd3 6789 . . . . . 6 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
2610, 25jaoi 853 . . . . 5 (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
274, 26sylbir 236 . . . 4 (((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
2827expcom 414 . . 3 ((¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}))
293, 28sylbi 218 . 2 ((𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}))
301, 2, 29mp2 9 1 (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500  c0 4295  cmpt 5143  dom cdm 5554  cfv 6354
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362
This theorem is referenced by: (None)
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