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Theorem fvimacnv 5225
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 4920 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
fvimacnv ((Fun 𝐹 A dom 𝐹) → ((𝐹A) BA (𝐹B)))

Proof of Theorem fvimacnv
StepHypRef Expression
1 funfvop 5222 . . . . 5 ((Fun 𝐹 A dom 𝐹) → ⟨A, (𝐹A)⟩ 𝐹)
2 funfvex 5135 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → (𝐹A) V)
3 opelcnvg 4458 . . . . . 6 (((𝐹A) V A dom 𝐹) → (⟨(𝐹A), A 𝐹 ↔ ⟨A, (𝐹A)⟩ 𝐹))
42, 3sylancom 397 . . . . 5 ((Fun 𝐹 A dom 𝐹) → (⟨(𝐹A), A 𝐹 ↔ ⟨A, (𝐹A)⟩ 𝐹))
51, 4mpbird 156 . . . 4 ((Fun 𝐹 A dom 𝐹) → ⟨(𝐹A), A 𝐹)
6 elimasng 4636 . . . . 5 (((𝐹A) V A dom 𝐹) → (A (𝐹 “ {(𝐹A)}) ↔ ⟨(𝐹A), A 𝐹))
72, 6sylancom 397 . . . 4 ((Fun 𝐹 A dom 𝐹) → (A (𝐹 “ {(𝐹A)}) ↔ ⟨(𝐹A), A 𝐹))
85, 7mpbird 156 . . 3 ((Fun 𝐹 A dom 𝐹) → A (𝐹 “ {(𝐹A)}))
9 snssg 3491 . . . . . . . 8 ((𝐹A) V → ((𝐹A) B ↔ {(𝐹A)} ⊆ B))
102, 9syl 14 . . . . . . 7 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) B ↔ {(𝐹A)} ⊆ B))
11 imass2 4644 . . . . . . 7 ({(𝐹A)} ⊆ B → (𝐹 “ {(𝐹A)}) ⊆ (𝐹B))
1210, 11syl6bi 152 . . . . . 6 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) B → (𝐹 “ {(𝐹A)}) ⊆ (𝐹B)))
1312imp 115 . . . . 5 (((Fun 𝐹 A dom 𝐹) (𝐹A) B) → (𝐹 “ {(𝐹A)}) ⊆ (𝐹B))
1413sseld 2938 . . . 4 (((Fun 𝐹 A dom 𝐹) (𝐹A) B) → (A (𝐹 “ {(𝐹A)}) → A (𝐹B)))
1514ex 108 . . 3 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) B → (A (𝐹 “ {(𝐹A)}) → A (𝐹B))))
168, 15mpid 37 . 2 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) BA (𝐹B)))
17 fvimacnvi 5224 . . . 4 ((Fun 𝐹 A (𝐹B)) → (𝐹A) B)
1817ex 108 . . 3 (Fun 𝐹 → (A (𝐹B) → (𝐹A) B))
1918adantr 261 . 2 ((Fun 𝐹 A dom 𝐹) → (A (𝐹B) → (𝐹A) B))
2016, 19impbid 120 1 ((Fun 𝐹 A dom 𝐹) → ((𝐹A) BA (𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  Vcvv 2551  wss 2911  {csn 3367  cop 3370  ccnv 4287  dom cdm 4288  cima 4291  Fun wfun 4839  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  funimass3  5226  elpreima  5229
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