Theorem elfvov2 7491

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	⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌))

Assertion

Ref	Expression
elfvov2	⊢ (𝜑 → 𝑅 ∈ V)

Proof of Theorem elfvov2

Step	Hyp	Ref	Expression
1		elfvov1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌))
2		n0i 4363	. . 3 ⊢ (𝑋 ∈ (𝑆‘𝑌) → ¬ (𝑆‘𝑌) = ∅)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ¬ (𝑆‘𝑌) = ∅)
4		elfvov1.s	. . . . 5 ⊢ 𝑆 = (𝐼𝑂𝑅)
5		elfvov1.o	. . . . . 6 ⊢ Rel dom 𝑂
6	5	ovprc2 7488	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅)
7	4, 6	eqtrid 2792	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
8	7	fveq1d 6922	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = (∅‘𝑌))
9		0fv 6964	. . 3 ⊢ (∅‘𝑌) = ∅
10	8, 9	eqtrdi 2796	. 2 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = ∅)
11	3, 10	nsyl2 141	1 ⊢ (𝜑 → 𝑅 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  dom cdm 5700  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  mhprcl  22170  mhpmpl  22171  mhpdeg  22172