ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvelima GIF version

Theorem fvelima 5441
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 4855 . . . 4 (𝐴 ∈ (𝐹𝐵) → (𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 𝑥𝐹𝐴))
21ibi 175 . . 3 (𝐴 ∈ (𝐹𝐵) → ∃𝑥𝐵 𝑥𝐹𝐴)
3 funbrfv 5428 . . . 4 (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹𝑥) = 𝐴))
43reximdv 2510 . . 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 1316  wcel 1465  wrex 2394   class class class wbr 3899  cima 4512  Fun wfun 5087  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by:  ssimaex  5450  ctssdccl  6964  suplocexprlemmu  7494  suplocexprlemloc  7497  ennnfonelemex  11854
  Copyright terms: Public domain W3C validator