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Theorem fvimacnv 6372
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 6010 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 6369 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
2 fvex 6239 . . . . . . 7 (𝐹𝐴) ∈ V
3 opelcnvg 5334 . . . . . . 7 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
42, 3mpan 706 . . . . . 6 (𝐴 ∈ dom 𝐹 → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
54adantl 481 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
61, 5mpbird 247 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹)
7 elimasng 5526 . . . . . 6 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
82, 7mpan 706 . . . . 5 (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
98adantl 481 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
106, 9mpbird 247 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐹 “ {(𝐹𝐴)}))
112snss 4348 . . . . 5 ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵)
12 imass2 5536 . . . . 5 ({(𝐹𝐴)} ⊆ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1311, 12sylbi 207 . . . 4 ((𝐹𝐴) ∈ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1413sseld 3635 . . 3 ((𝐹𝐴) ∈ 𝐵 → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) → 𝐴 ∈ (𝐹𝐵)))
1510, 14syl5com 31 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
16 fvimacnvi 6371 . . . 4 ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
1716ex 449 . . 3 (Fun 𝐹 → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1817adantr 480 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1915, 18impbid 202 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  Vcvv 3231  wss 3607  {csn 4210  cop 4216  ccnv 5142  dom cdm 5143  cima 5146  Fun wfun 5920  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  funimass3  6373  elpreima  6377  iinpreima  6385  isr0  21588  rnelfmlem  21803  rnelfm  21804  fmfnfmlem2  21806  fmfnfmlem4  21808  fmfnfm  21809  metustid  22406  metustsym  22407  metustexhalf  22408  xppreima  29577  dstfrvel  30663  ballotlemrv  30709  grpokerinj  33822  diaintclN  36664  dibintclN  36773  dihintcl  36950  arearect  38118  areaquad  38119
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