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Theorem fvimacnv 6818
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 6433 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 6815 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
2 fvex 6679 . . . . . . 7 (𝐹𝐴) ∈ V
3 opelcnvg 5749 . . . . . . 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 5952 . . . . . 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 4716 . . . . 5 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
12 imass2 5962 . . . . 5 ({(𝐹𝐴)} ⊆ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1311, 12sylbi 218 . . . 4 ((𝐹𝐴) ∈ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1413sseld 3969 . . 3 ((𝐹𝐴) ∈ 𝐵 → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) → 𝐴 ∈ (𝐹𝐵)))
1510, 14syl5com 31 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
16 fvimacnvi 6817 . . . 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 2106  Vcvv 3499  wss 3939  {csn 4563  cop 4569  ccnv 5552  dom cdm 5553  cima 5556  Fun wfun 6345  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359
This theorem is referenced by:  funimass3  6819  elpreima  6823  iinpreima  6832  isr0  22261  rnelfmlem  22476  rnelfm  22477  fmfnfmlem2  22479  fmfnfmlem4  22481  fmfnfm  22482  metustid  23079  metustsym  23080  metustexhalf  23081  xppreima  30310  dstfrvel  31618  ballotlemrv  31664  bj-fvimacnv0  34444  grpokerinj  35040  diaintclN  38062  dibintclN  38171  dihintcl  38348  arearect  39684  areaquad  39685
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