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Theorem fnmptd 14178
Description: The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
Hypotheses
Ref Expression
fnmptd.def (𝜑𝐹 = (𝑥𝐴𝐵))
fnmptd.ex ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
fnmptd (𝜑𝐹 Fn 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fnmptd
StepHypRef Expression
1 fnmptd.ex . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
21ralrimiva 2550 . . 3 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
3 eqid 2177 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43fnmpt 5337 . . 3 (∀𝑥𝐴 𝐵𝑉 → (𝑥𝐴𝐵) Fn 𝐴)
52, 4syl 14 . 2 (𝜑 → (𝑥𝐴𝐵) Fn 𝐴)
6 fnmptd.def . . 3 (𝜑𝐹 = (𝑥𝐴𝐵))
76fneq1d 5301 . 2 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴))
85, 7mpbird 167 1 (𝜑𝐹 Fn 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  cmpt 4061   Fn wfn 5206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-fun 5213  df-fn 5214
This theorem is referenced by: (None)
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