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Theorem elfvov2 7473
Description: Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 4-Aug-2025.)
Hypotheses
Ref Expression
elfvov1.o Rel dom 𝑂
elfvov1.s 𝑆 = (𝐼𝑂𝑅)
elfvov1.x (𝜑𝑋 ∈ (𝑆𝑌))
Assertion
Ref Expression
elfvov2 (𝜑𝑅 ∈ V)

Proof of Theorem elfvov2
StepHypRef Expression
1 elfvov1.x . . 3 (𝜑𝑋 ∈ (𝑆𝑌))
2 n0i 4345 . . 3 (𝑋 ∈ (𝑆𝑌) → ¬ (𝑆𝑌) = ∅)
31, 2syl 17 . 2 (𝜑 → ¬ (𝑆𝑌) = ∅)
4 elfvov1.s . . . . 5 𝑆 = (𝐼𝑂𝑅)
5 elfvov1.o . . . . . 6 Rel dom 𝑂
65ovprc2 7470 . . . . 5 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅)
74, 6eqtrid 2786 . . . 4 𝑅 ∈ V → 𝑆 = ∅)
87fveq1d 6908 . . 3 𝑅 ∈ V → (𝑆𝑌) = (∅‘𝑌))
9 0fv 6950 . . 3 (∅‘𝑌) = ∅
108, 9eqtrdi 2790 . 2 𝑅 ∈ V → (𝑆𝑌) = ∅)
113, 10nsyl2 141 1 (𝜑𝑅 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  dom cdm 5688  Rel wrel 5693  cfv 6562  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-dm 5698  df-iota 6515  df-fv 6570  df-ov 7433
This theorem is referenced by:  ismhp  22161  mhprcl  22164
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