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Theorem fnmptif 45272
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Hypotheses
Ref Expression
fnmptif.1 𝑥𝐴
fnmptif.2 𝐵 ∈ V
fnmptif.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fnmptif 𝐹 Fn 𝐴

Proof of Theorem fnmptif
StepHypRef Expression
1 fnmptif.2 . . . 4 𝐵 ∈ V
21rgenw 3065 . . 3 𝑥𝐴 𝐵 ∈ V
3 fnmptif.1 . . . 4 𝑥𝐴
43mptfnf 6703 . . 3 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
52, 4mpbi 230 . 2 (𝑥𝐴𝐵) Fn 𝐴
6 fnmptif.3 . . 3 𝐹 = (𝑥𝐴𝐵)
76fneq1i 6665 . 2 (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴)
85, 7mpbir 231 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  Vcvv 3480  cmpt 5225   Fn wfn 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563  df-fn 6564
This theorem is referenced by:  dmmptif  45273
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