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Theorem elfvov2 7474
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 4340 . . 3 (𝑋 ∈ (𝑆𝑌) → ¬ (𝑆𝑌) = ∅)
31, 2syl 17 . 2 (𝜑 → ¬ (𝑆𝑌) = ∅)
4 elfvov1.s . . . . 5 𝑆 = (𝐼𝑂𝑅)
5 elfvov1.o . . . . . 6 Rel dom 𝑂
65ovprc2 7471 . . . . 5 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅)
74, 6eqtrid 2789 . . . 4 𝑅 ∈ V → 𝑆 = ∅)
87fveq1d 6908 . . 3 𝑅 ∈ V → (𝑆𝑌) = (∅‘𝑌))
9 0fv 6950 . . 3 (∅‘𝑌) = ∅
108, 9eqtrdi 2793 . 2 𝑅 ∈ V → (𝑆𝑌) = ∅)
113, 10nsyl2 141 1 (𝜑𝑅 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  dom cdm 5685  Rel wrel 5690  cfv 6561  (class class class)co 7431
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-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-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-iota 6514  df-fv 6569  df-ov 7434
This theorem is referenced by:  ismhp  22144  mhprcl  22147
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