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Theorem fvmptrabdm 41817
Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6471. (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 pm2.1 432 . 2 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹)
3 imor 427 . . 3 ((𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) ↔ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹))
4 ordir 945 . . . . 5 (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) ↔ ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)))
5 ndmfv 6379 . . . . . . 7 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
6 ndmfv 6379 . . . . . . . . 9 𝑌 ∈ dom 𝐺 → (𝐺𝑌) = ∅)
76rabeqdv 3334 . . . . . . . 8 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8 rab0 4098 . . . . . . . 8 {𝑦 ∈ ∅ ∣ 𝜓} = ∅
97, 8syl6req 2811 . . . . . . 7 𝑌 ∈ dom 𝐺 → ∅ = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
105, 9sylan9eq 2814 . . . . . 6 ((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
11 fvmptrabdm.f . . . . . . . 8 𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑})
1211a1i 11 . . . . . . 7 (𝑋 ∈ dom 𝐹𝐹 = (𝑥𝑉 ↦ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑}))
13 fvmptrabdm.r . . . . . . . . 9 (𝑥 = 𝑋 → (𝜑𝜓))
1413rabbidv 3329 . . . . . . . 8 (𝑥 = 𝑋 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
1514adantl 473 . . . . . . 7 ((𝑋 ∈ dom 𝐹𝑥 = 𝑋) → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
1611dmmpt 5791 . . . . . . . . . 10 dom 𝐹 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V}
17 rabid2 3257 . . . . . . . . . . 11 (𝑉 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥𝑉 {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V)
18 fvex 6362 . . . . . . . . . . . . 13 (𝐺𝑌) ∈ V
1918rabex 4964 . . . . . . . . . . . 12 {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝑥𝑉 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V)
2117, 20mprgbir 3065 . . . . . . . . . 10 𝑉 = {𝑥𝑉 ∣ {𝑦 ∈ (𝐺𝑌) ∣ 𝜑} ∈ V}
2216, 21eqtr4i 2785 . . . . . . . . 9 dom 𝐹 = 𝑉
2322eleq2i 2831 . . . . . . . 8 (𝑋 ∈ dom 𝐹𝑋𝑉)
2423biimpi 206 . . . . . . 7 (𝑋 ∈ dom 𝐹𝑋𝑉)
2518rabex 4964 . . . . . . . 8 {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} ∈ V
2625a1i 11 . . . . . . 7 (𝑋 ∈ dom 𝐹 → {𝑦 ∈ (𝐺𝑌) ∣ 𝜓} ∈ V)
2712, 15, 24, 26fvmptd 6450 . . . . . 6 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
2810, 27jaoi 393 . . . . 5 (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
294, 28sylbir 225 . . . 4 (((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹)) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓})
3029expcom 450 . . 3 ((¬ 𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}))
313, 30sylbi 207 . 2 ((𝑌 ∈ dom 𝐺𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐹) → (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}))
321, 2, 31mp2 9 1 (𝐹𝑋) = {𝑦 ∈ (𝐺𝑌) ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  c0 4058  cmpt 4881  dom cdm 5266  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057
This theorem is referenced by: (None)
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