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Theorem dfimafn2 6833
Description: Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
dfimafn2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfimafn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfimafn 6832 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
2 iunab 4981 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
31, 2eqtr4di 2796 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦})
4 df-sn 4562 . . . . 5 {(𝐹𝑥)} = {𝑦𝑦 = (𝐹𝑥)}
5 eqcom 2745 . . . . . 6 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
65abbii 2808 . . . . 5 {𝑦𝑦 = (𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
74, 6eqtri 2766 . . . 4 {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
87a1i 11 . . 3 (𝑥𝐴 → {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦})
98iuneq2i 4945 . 2 𝑥𝐴 {(𝐹𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦}
103, 9eqtr4di 2796 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {cab 2715  wrex 3065  wss 3887  {csn 4561   ciun 4924  dom cdm 5589  cima 5592  Fun wfun 6427  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441
This theorem is referenced by:  uniiccdif  24742
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