Theorem elfvov1 7429

Description: Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 18-May-2025.)

Hypotheses

Ref	Expression
elfvov1.o	⊢ Rel dom 𝑂
elfvov1.s	⊢ 𝑆 = (𝐼𝑂𝑅)
elfvov1.x	⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌))

Assertion

Ref	Expression
elfvov1	⊢ (𝜑 → 𝐼 ∈ V)

Proof of Theorem elfvov1

Step	Hyp	Ref	Expression
1		elfvov1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑆‘𝑌))
2		n0i 4303	. . 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	ovprc1 7426	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼𝑂𝑅) = ∅)
7	4, 6	eqtrid 2776	. . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑆 = ∅)
8	7	fveq1d 6860	. . 3 ⊢ (¬ 𝐼 ∈ V → (𝑆‘𝑌) = (∅‘𝑌))
9		0fv 6902	. . 3 ⊢ (∅‘𝑌) = ∅
10	8, 9	eqtrdi 2780	. 2 ⊢ (¬ 𝐼 ∈ V → (𝑆‘𝑌) = ∅)
11	3, 10	nsyl2 141	1 ⊢ (𝜑 → 𝐼 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∅c0 4296  dom cdm 5638  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-iota 6464  df-fv 6519  df-ov 7390

This theorem is referenced by:  ismhp  22027  mhprcl  22030  mhpmulcl  22036  mhppwdeg  22037  mhpaddcl  22038  mhpinvcl  22039  mhpvscacl  22041  mhpind  42582  evlsmhpvvval  42583  mhphf2  42586  mhphf3  42587