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

Theorem dfimafn 5517
Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
dfimafn ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem dfimafn
StepHypRef Expression
1 dfima2 4930 . 2 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
2 ssel 3122 . . . . . 6 (𝐴 ⊆ dom 𝐹 → (𝑥𝐴𝑥 ∈ dom 𝐹))
3 funbrfvb 5511 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
43ex 114 . . . . . 6 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦)))
52, 4syl9r 73 . . . . 5 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))))
65imp31 254 . . . 4 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
76rexbidva 2454 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐴 𝑥𝐹𝑦))
87abbidv 2275 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦})
91, 8eqtr4id 2209 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  {cab 2143  wrex 2436  wss 3102   class class class wbr 3965  dom cdm 4586  cima 4589  Fun wfun 5164  cfv 5170
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178
This theorem is referenced by:  dfimafn2  5518  fvelimab  5524
  Copyright terms: Public domain W3C validator