Theorem elfvov2 7446

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 4315	. . 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 7443	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅)
7	4, 6	eqtrid 2782	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
8	7	fveq1d 6877	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = (∅‘𝑌))
9		0fv 6919	. . 3 ⊢ (∅‘𝑌) = ∅
10	8, 9	eqtrdi 2786	. 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 3459  ∅c0 4308  dom cdm 5654  Rel wrel 5659  ‘cfv 6530  (class class class)co 7403

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-dm 5664  df-iota 6483  df-fv 6538  df-ov 7406

This theorem is referenced by:  ismhp  22076  mhprcl  22079