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Theorem fvelima 5369
Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
fvelima ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹𝑥) = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fvelima
StepHypRef Expression
1 elimag 4791 . . . 4 (𝐴 ∈ (𝐹𝐵) → (𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 𝑥𝐹𝐴))
21ibi 175 . . 3 (𝐴 ∈ (𝐹𝐵) → ∃𝑥𝐵 𝑥𝐹𝐴)
3 funbrfv 5356 . . . 4 (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹𝑥) = 𝐴))
43reximdv 2475 . . 3 (Fun 𝐹 → (∃𝑥𝐵 𝑥𝐹𝐴 → ∃𝑥𝐵 (𝐹𝑥) = 𝐴))
52, 4syl5 32 . 2 (Fun 𝐹 → (𝐴 ∈ (𝐹𝐵) → ∃𝑥𝐵 (𝐹𝑥) = 𝐴))
65imp 123 1 ((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → ∃𝑥𝐵 (𝐹𝑥) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wcel 1439  wrex 2361   class class class wbr 3851  cima 4455  Fun wfun 5022  cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fv 5036
This theorem is referenced by:  ssimaex  5378
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