Theorem elfvov2 7466

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 4296	. . 3 ⊢ (𝑋 ∈ (𝑆‘𝑌) → ¬ (𝑆‘𝑌) = ∅)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → ¬ (𝑆‘𝑌) = ∅)
4		elfvov1.s	. . . . 5 ⊢ 𝑆 = (𝐼𝑂𝑅)
5		elfvov1.o	. . . . . 6 ⊢ Rel dom 𝑂
6	5	ovprc2 7463	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐼𝑂𝑅) = ∅)
7	4, 6	eqtrid 2813	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑆 = ∅)
8	7	fveq1d 6890	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = (∅‘𝑌))
9		0fv 6929	. . 3 ⊢ (∅‘𝑌) = ∅
10	8, 9	eqtrdi 2817	. 2 ⊢ (¬ 𝑅 ∈ V → (𝑆‘𝑌) = ∅)
11	3, 10	nsyl2 142	1 ⊢ (𝜑 → 𝑅 ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ∅c0 4289  dom cdm 5666  Rel wrel 5671  ‘cfv 6543  (class class class)co 7423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-dm 5676  df-iota 6499  df-fv 6551  df-ov 7426

This theorem is used by:  ismhp  22333  mhprcl  22336