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Theorem fnmptif 45211
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 3063 . . 3 𝑥𝐴 𝐵 ∈ V
3 fnmptif.1 . . . 4 𝑥𝐴
43mptfnf 6704 . . 3 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
52, 4mpbi 230 . 2 (𝑥𝐴𝐵) Fn 𝐴
6 fnmptif.3 . . 3 𝐹 = (𝑥𝐴𝐵)
76fneq1i 6666 . 2 (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴)
85, 7mpbir 231 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wnfc 2888  wral 3059  Vcvv 3478  cmpt 5231   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566
This theorem is referenced by:  dmmptif  45212
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