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Theorem elfvmptrab 7045
Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elfvmptrab.f 𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})
elfvmptrab.v (𝑋𝑉𝑀 ∈ V)
Assertion
Ref Expression
elfvmptrab (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑦,𝑌
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑦)   𝑌(𝑥)

Proof of Theorem elfvmptrab
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 elfvmptrab.f . . . 4 𝐹 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})
2 csbconstg 3918 . . . . . . 7 (𝑥𝑉𝑥 / 𝑚𝑀 = 𝑀)
32eqcomd 2743 . . . . . 6 (𝑥𝑉𝑀 = 𝑥 / 𝑚𝑀)
4 rabeq 3451 . . . . . 6 (𝑀 = 𝑥 / 𝑚𝑀 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
53, 4syl 17 . . . . 5 (𝑥𝑉 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
65mpteq2ia 5245 . . . 4 (𝑥𝑉 ↦ {𝑦𝑀𝜑}) = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
71, 6eqtri 2765 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
8 csbconstg 3918 . . . 4 (𝑋𝑉𝑋 / 𝑚𝑀 = 𝑀)
9 elfvmptrab.v . . . 4 (𝑋𝑉𝑀 ∈ V)
108, 9eqeltrd 2841 . . 3 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
117, 10elfvmptrab1w 7043 . 2 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
128eleq2d 2827 . . . 4 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1312biimpd 229 . . 3 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1413imdistani 568 . 2 ((𝑋𝑉𝑌𝑋 / 𝑚𝑀) → (𝑋𝑉𝑌𝑀))
1511, 14syl 17 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  csb 3899  cmpt 5225  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by: (None)
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