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Theorem elfvmptrab 6772
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 3879 . . . . . . 7 (𝑥𝑉𝑥 / 𝑚𝑀 = 𝑀)
32eqcomd 2826 . . . . . 6 (𝑥𝑉𝑀 = 𝑥 / 𝑚𝑀)
4 rabeq 3462 . . . . . 6 (𝑀 = 𝑥 / 𝑚𝑀 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
53, 4syl 17 . . . . 5 (𝑥𝑉 → {𝑦𝑀𝜑} = {𝑦𝑥 / 𝑚𝑀𝜑})
65mpteq2ia 5133 . . . 4 (𝑥𝑉 ↦ {𝑦𝑀𝜑}) = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
71, 6eqtri 2843 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
8 csbconstg 3879 . . . 4 (𝑋𝑉𝑋 / 𝑚𝑀 = 𝑀)
9 elfvmptrab.v . . . 4 (𝑋𝑉𝑀 ∈ V)
108, 9eqeltrd 2911 . . 3 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
117, 10elfvmptrab1w 6770 . 2 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
128eleq2d 2896 . . . 4 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1312biimpd 231 . . 3 (𝑋𝑉 → (𝑌𝑋 / 𝑚𝑀𝑌𝑀))
1413imdistani 571 . 2 ((𝑋𝑉𝑌𝑋 / 𝑚𝑀) → (𝑋𝑉𝑌𝑀))
1511, 14syl 17 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  {crab 3129  Vcvv 3473  csb 3860  cmpt 5122  cfv 6331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fv 6339
This theorem is referenced by: (None)
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