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Theorem fvimacnv 5528
 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 5196 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 5525 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
2 funfvex 5431 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
3 opelcnvg 4714 . . . . . 6 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
42, 3sylancom 416 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
51, 4mpbird 166 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹)
6 elimasng 4902 . . . . 5 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
72, 6sylancom 416 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
85, 7mpbird 166 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐹 “ {(𝐹𝐴)}))
9 snssg 3651 . . . . . . . 8 ((𝐹𝐴) ∈ V → ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵))
102, 9syl 14 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵))
11 imass2 4910 . . . . . . 7 ({(𝐹𝐴)} ⊆ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1210, 11syl6bi 162 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵)))
1312imp 123 . . . . 5 (((Fun 𝐹𝐴 ∈ dom 𝐹) ∧ (𝐹𝐴) ∈ 𝐵) → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1413sseld 3091 . . . 4 (((Fun 𝐹𝐴 ∈ dom 𝐹) ∧ (𝐹𝐴) ∈ 𝐵) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) → 𝐴 ∈ (𝐹𝐵)))
1514ex 114 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) → 𝐴 ∈ (𝐹𝐵))))
168, 15mpid 42 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
17 fvimacnvi 5527 . . . 4 ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
1817ex 114 . . 3 (Fun 𝐹 → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1918adantr 274 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
2016, 19impbid 128 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  Vcvv 2681   ⊆ wss 3066  {csn 3522  ⟨cop 3525  ◡ccnv 4533  dom cdm 4534   “ cima 4537  Fun wfun 5112  ‘cfv 5118 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126 This theorem is referenced by:  funimass3  5529  elpreima  5532  fisumss  11154
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