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

Theorem elfvmptrab 6980
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 3878 . . . . . . 7 (𝑥𝑉𝑥 / 𝑚𝑀 = 𝑀)
32eqcomd 2739 . . . . . 6 (𝑥𝑉𝑀 = 𝑥 / 𝑚𝑀)
4 rabeq 3420 . . . . . 6 (𝑀 = 𝑥 / 𝑚𝑀 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
53, 4syl 17 . . . . 5 (𝑥𝑉 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
65mpteq2ia 5212 . . . 4 (𝑥𝑉 ↦ {𝑦𝑀𝜑}) = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
71, 6eqtri 2761 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
8 csbconstg 3878 . . . 4 (𝑋𝑉𝑋 / 𝑚𝑀 = 𝑀)
9 elfvmptrab.v . . . 4 (𝑋𝑉𝑀 ∈ V)
108, 9eqeltrd 2834 . . 3 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
117, 10elfvmptrab1w 6978 . 2 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
128eleq2d 2820 . . . 4 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1312biimpd 228 . . 3 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1413imdistani 570 . 2 ((𝑋𝑉𝑌𝑋 / 𝑚𝑀) → (𝑋𝑉𝑌𝑀))
1511, 14syl 17 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3447  csb 3859  cmpt 5192  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator