Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elfvmptrab Structured version   Visualization version   GIF version

Theorem elfvmptrab 6568
 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 3763 . . . . . . 7 (𝑥𝑉𝑥 / 𝑚𝑀 = 𝑀)
32eqcomd 2783 . . . . . 6 (𝑥𝑉𝑀 = 𝑥 / 𝑚𝑀)
4 rabeq 3388 . . . . . 6 (𝑀 = 𝑥 / 𝑚𝑀 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
53, 4syl 17 . . . . 5 (𝑥𝑉 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
65mpteq2ia 4975 . . . 4 (𝑥𝑉 ↦ {𝑦𝑀𝜑}) = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
71, 6eqtri 2801 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
8 csbconstg 3763 . . . 4 (𝑋𝑉𝑋 / 𝑚𝑀 = 𝑀)
9 elfvmptrab.v . . . 4 (𝑋𝑉𝑀 ∈ V)
108, 9eqeltrd 2858 . . 3 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
117, 10elfvmptrab1 6567 . 2 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
128eleq2d 2844 . . . 4 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1312biimpd 221 . . 3 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1413imdistani 564 . 2 ((𝑋𝑉𝑌𝑋 / 𝑚𝑀) → (𝑋𝑉𝑌𝑀))
1511, 14syl 17 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2106  {crab 3093  Vcvv 3397  ⦋csb 3750   ↦ cmpt 4965  ‘cfv 6135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator