MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elfvov1 Structured version   Visualization version   GIF version

Theorem elfvov1 7473
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 (𝜑𝑋 ∈ (𝑆𝑌))
Assertion
Ref Expression
elfvov1 (𝜑𝐼 ∈ V)

Proof of Theorem elfvov1
StepHypRef Expression
1 elfvov1.x . . 3 (𝜑𝑋 ∈ (𝑆𝑌))
2 n0i 4346 . . 3 (𝑋 ∈ (𝑆𝑌) → ¬ (𝑆𝑌) = ∅)
31, 2syl 17 . 2 (𝜑 → ¬ (𝑆𝑌) = ∅)
4 elfvov1.s . . . . 5 𝑆 = (𝐼𝑂𝑅)
5 elfvov1.o . . . . . 6 Rel dom 𝑂
65ovprc1 7470 . . . . 5 𝐼 ∈ V → (𝐼𝑂𝑅) = ∅)
74, 6eqtrid 2787 . . . 4 𝐼 ∈ V → 𝑆 = ∅)
87fveq1d 6909 . . 3 𝐼 ∈ V → (𝑆𝑌) = (∅‘𝑌))
9 0fv 6951 . . 3 (∅‘𝑌) = ∅
108, 9eqtrdi 2791 . 2 𝐼 ∈ V → (𝑆𝑌) = ∅)
113, 10nsyl2 141 1 (𝜑𝐼 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  dom cdm 5689  Rel wrel 5694  cfv 6563  (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 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  ismhp  22162  mhprcl  22165  mhpmulcl  22171  mhppwdeg  22172  mhpaddcl  22173  mhpinvcl  22174  mhpvscacl  22176  mhpind  42581  evlsmhpvvval  42582  mhphf2  42585  mhphf3  42586
  Copyright terms: Public domain W3C validator