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Theorem dfimafn 6784
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 5940 . 2 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
2 ssel 3902 . . . . . 6 (𝐴 ⊆ dom 𝐹 → (𝑥𝐴𝑥 ∈ dom 𝐹))
3 funbrfvb 6776 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
43ex 416 . . . . . 6 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦)))
52, 4syl9r 78 . . . . 5 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))))
65imp31 421 . . . 4 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
76rexbidva 3222 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐴 𝑥𝐹𝑦))
87abbidv 2808 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦})
91, 8eqtr4id 2798 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  {cab 2715  wrex 3063  wss 3875   class class class wbr 5062  dom cdm 5560  cima 5563  Fun wfun 6383  cfv 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-br 5063  df-opab 5125  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-iota 6347  df-fun 6391  df-fn 6392  df-fv 6397
This theorem is referenced by:  dfimafn2  6785  fvelimab  6793  cshimadifsn  14407  cshimadifsn0  14408  ushgredgedg  27330  ushgredgedgloop  27332  curry2ima  30774  intimafv  30776  fnrelpredd  32787  poimirlem26  35553  poimirlem27  35554  f1oresf1o  44469  imasetpreimafvbijlemfo  44545
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