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Theorem dfaimafn2 43242
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6722. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfaimafn2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfaimafn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfaimafn 43241 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
2 iunab 4966 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦}
31, 2syl6eqr 2871 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
4 df-sn 4558 . . . . 5 {(𝐹'''𝑥)} = {𝑦𝑦 = (𝐹'''𝑥)}
5 eqcom 2825 . . . . . 6 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
65abbii 2883 . . . . 5 {𝑦𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
74, 6eqtri 2841 . . . 4 {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
87a1i 11 . . 3 (𝑥𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
98iuneq2i 4931 . 2 𝑥𝐴 {(𝐹'''𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
103, 9syl6eqr 2871 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  wss 3933  {csn 4557   ciun 4910  dom cdm 5548  cima 5551  Fun wfun 6342  '''cafv 43193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-aiota 43162  df-dfat 43195  df-afv 43196
This theorem is referenced by: (None)
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