Theorem elfvov1 7402

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 4293	. . 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 7399	. . . . 5 ⊢ (¬ 𝐼 ∈ V → (𝐼𝑂𝑅) = ∅)
7	4, 6	eqtrid 2784	. . . 4 ⊢ (¬ 𝐼 ∈ V → 𝑆 = ∅)
8	7	fveq1d 6837	. . 3 ⊢ (¬ 𝐼 ∈ V → (𝑆‘𝑌) = (∅‘𝑌))
9		0fv 6876	. . 3 ⊢ (∅‘𝑌) = ∅
10	8, 9	eqtrdi 2788	. 2 ⊢ (¬ 𝐼 ∈ V → (𝑆‘𝑌) = ∅)
11	3, 10	nsyl2 141	1 ⊢ (𝜑 → 𝐼 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3441  ∅c0 4286  dom cdm 5625  Rel wrel 5630  ‘cfv 6493  (class class class)co 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-dm 5635  df-iota 6449  df-fv 6501  df-ov 7363

This theorem is referenced by:  ismhp  22087  mhprcl  22090  mhpmulcl  22096  mhppwdeg  22097  mhpaddcl  22098  mhpinvcl  22099  mhpvscacl  22101  mhpind  42873  evlsmhpvvval  42874  mhphf2  42877  mhphf3  42878