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Theorem mptfnd 40375
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
mptfnd.1 𝑥𝐴
mptfnd.2 𝑥𝜑
mptfnd.3 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
mptfnd (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)

Proof of Theorem mptfnd
StepHypRef Expression
1 mptfnd.2 . . 3 𝑥𝜑
2 mptfnd.3 . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ex 403 . . . 4 (𝜑 → (𝑥𝐴𝐵𝑉))
4 elex 3414 . . . 4 (𝐵𝑉𝐵 ∈ V)
53, 4syl6 35 . . 3 (𝜑 → (𝑥𝐴𝐵 ∈ V))
61, 5ralrimi 3139 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ V)
7 mptfnd.1 . . 3 𝑥𝐴
87mptfnf 6263 . 2 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
96, 8sylib 210 1 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wnf 1827  wcel 2107  wnfc 2919  wral 3090  Vcvv 3398  cmpt 4967   Fn wfn 6132
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 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
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 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-fun 6139  df-fn 6140
This theorem is referenced by:  smflimsuplem2  41964
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