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Theorem fnmptif 44671
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 3062 . . 3 𝑥𝐴 𝐵 ∈ V
3 fnmptif.1 . . . 4 𝑥𝐴
43mptfnf 6695 . . 3 (∀𝑥𝐴 𝐵 ∈ V ↔ (𝑥𝐴𝐵) Fn 𝐴)
52, 4mpbi 229 . 2 (𝑥𝐴𝐵) Fn 𝐴
6 fnmptif.3 . . 3 𝐹 = (𝑥𝐴𝐵)
76fneq1i 6656 . 2 (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐵) Fn 𝐴)
85, 7mpbir 230 1 𝐹 Fn 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wnfc 2879  wral 3058  Vcvv 3473  cmpt 5235   Fn wfn 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-fun 6555  df-fn 6556
This theorem is referenced by:  dmmptif  44672
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