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Theorem fvimacnv 5655
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 5316 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 5652 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
2 funfvex 5554 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ V)
3 opelcnvg 4828 . . . . . 6 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
42, 3sylancom 420 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹 ↔ ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹))
51, 4mpbird 167 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹)
6 elimasng 5017 . . . . 5 (((𝐹𝐴) ∈ V ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
72, 6sylancom 420 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) ↔ ⟨(𝐹𝐴), 𝐴⟩ ∈ 𝐹))
85, 7mpbird 167 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐹 “ {(𝐹𝐴)}))
9 snssg 3744 . . . . . . . 8 ((𝐹𝐴) ∈ V → ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵))
102, 9syl 14 . . . . . . 7 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 ↔ {(𝐹𝐴)} ⊆ 𝐵))
11 imass2 5025 . . . . . . 7 ({(𝐹𝐴)} ⊆ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1210, 11biimtrdi 163 . . . . . 6 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵 → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵)))
1312imp 124 . . . . 5 (((Fun 𝐹𝐴 ∈ dom 𝐹) ∧ (𝐹𝐴) ∈ 𝐵) → (𝐹 “ {(𝐹𝐴)}) ⊆ (𝐹𝐵))
1413sseld 3169 . . . 4 (((Fun 𝐹𝐴 ∈ dom 𝐹) ∧ (𝐹𝐴) ∈ 𝐵) → (𝐴 ∈ (𝐹 “ {(𝐹𝐴)}) → 𝐴 ∈ (𝐹𝐵)))
1514ex 115 . . 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 5654 . . . 4 ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
1817ex 115 . . 3 (Fun 𝐹 → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
1918adantr 276 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝐵) → (𝐹𝐴) ∈ 𝐵))
2016, 19impbid 129 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2160  Vcvv 2752  wss 3144  {csn 3610  cop 3613  ccnv 4646  dom cdm 4647  cima 4650  Fun wfun 5232  cfv 5238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246
This theorem is referenced by:  funimass3  5656  elpreima  5659  fisumss  11441  psrbaglesuppg  13975
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