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Theorem dfaimafn 47177
Description: Alternate definition of the image of a function, analogous to dfimafn 6971. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfaimafn ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem dfaimafn
StepHypRef Expression
1 dfima2 6080 . 2 (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦}
2 ssel 3977 . . . . . 6 (𝐴 ⊆ dom 𝐹 → (𝑥𝐴𝑥 ∈ dom 𝐹))
3 funbrafvb 47168 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))
43ex 412 . . . . . 6 (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦)))
52, 4syl9r 78 . . . . 5 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥𝐴 → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))))
65imp31 417 . . . 4 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑥𝐴) → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))
76rexbidva 3177 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑥𝐴 (𝐹'''𝑥) = 𝑦 ↔ ∃𝑥𝐴 𝑥𝐹𝑦))
87abbidv 2808 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑥𝐹𝑦})
91, 8eqtr4id 2796 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  wss 3951   class class class wbr 5143  dom cdm 5685  cima 5688  Fun wfun 6555  '''cafv 47129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132
This theorem is referenced by:  dfaimafn2  47178
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