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Theorem fvimacnv 6815
Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6430 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
fvimacnv ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))

Proof of Theorem fvimacnv
StepHypRef Expression
1 funfvop 6812 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
2 fvex 6676 . . . . . . 7 (𝐹𝐴) ∈ V
3 opelcnvg 5744 . . . . . . 7 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
42, 3mpan 686 . . . . . 6 (𝐴 ∈ dom 𝐹 → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
54adantl 482 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
61, 5mpbird 258 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹)
7 elimasng 5948 . . . . . 6 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
82, 7mpan 686 . . . . 5 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
98adantl 482 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
106, 9mpbird 258 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐹 “ {(𝐹𝐴)}))
112snss 4710 . . . . 5 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
12 imass2 5958 . . . . 5 ({(𝐹𝐴)} ⊆ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1311, 12sylbi 218 . . . 4 ((𝐹𝐴) ∈ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1413sseld 3963 . . 3 ((𝐹𝐴) ∈ 𝐵 → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) → 𝐴 ∈ (𝐹𝐵)))
1510, 14syl5com 31 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
16 fvimacnvi 6814 . . . 4 ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
1716ex 413 . . 3 (Fun 𝐹 → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1817adantr 481 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1915, 18impbid 213 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  Vcvv 3492  wss 3933  {csn 4557  cop 4563  ccnv 5547  dom cdm 5548  cima 5551  Fun wfun 6342  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  funimass3  6816  elpreima  6820  iinpreima  6829  isr0  22273  rnelfmlem  22488  rnelfm  22489  fmfnfmlem2  22491  fmfnfmlem4  22493  fmfnfm  22494  metustid  23091  metustsym  23092  metustexhalf  23093  xppreima  30322  dstfrvel  31630  ballotlemrv  31676  bj-fvimacnv0  34456  grpokerinj  35052  diaintclN  38074  dibintclN  38183  dihintcl  38360  arearect  39700  areaquad  39701
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