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	⊢ (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 4340	. . 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 7471	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅)
7	4, 6	eqtrid 2789	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
8	7	fveq1d 6908	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = (∅‘𝑌))
9		0fv 6950	. . 3 ⊢ (∅‘𝑌) = ∅
10	8, 9	eqtrdi 2793	. 2 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = ∅)
11	3, 10	nsyl2 141	1 ⊢ (𝜑 → 𝑅 ∈ V)

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