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Theorem elfvmptrab 6450
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 3695 . . . . . . 7 (𝑥𝑉𝑥 / 𝑚𝑀 = 𝑀)
32eqcomd 2777 . . . . . 6 (𝑥𝑉𝑀 = 𝑥 / 𝑚𝑀)
4 rabeq 3342 . . . . . 6 (𝑀 = 𝑥 / 𝑚𝑀 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
53, 4syl 17 . . . . 5 (𝑥𝑉 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
65mpteq2ia 4875 . . . 4 (𝑥𝑉 ↦ {𝑦𝑀𝜑}) = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
71, 6eqtri 2793 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
8 csbconstg 3695 . . . 4 (𝑋𝑉𝑋 / 𝑚𝑀 = 𝑀)
9 elfvmptrab.v . . . 4 (𝑋𝑉𝑀 ∈ V)
108, 9eqeltrd 2850 . . 3 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
117, 10elfvmptrab1 6449 . 2 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
128eleq2d 2836 . . . 4 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1312biimpd 219 . . 3 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1413imdistani 558 . 2 ((𝑋𝑉𝑌𝑋 / 𝑚𝑀) → (𝑋𝑉𝑌𝑀))
1511, 14syl 17 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  csb 3682  cmpt 4864  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038
This theorem is referenced by: (None)
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