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Theorem dfaimafn2 44112
 Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6717. (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 44111 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
2 iunab 4940 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦}
31, 2eqtr4di 2811 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
4 df-sn 4523 . . . . 5 {(𝐹'''𝑥)} = {𝑦𝑦 = (𝐹'''𝑥)}
5 eqcom 2765 . . . . . 6 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
65abbii 2823 . . . . 5 {𝑦𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
74, 6eqtri 2781 . . . 4 {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
87a1i 11 . . 3 (𝑥𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
98iuneq2i 4904 . 2 𝑥𝐴 {(𝐹'''𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
103, 9eqtr4di 2811 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ⊆ wss 3858  {csn 4522  ∪ ciun 4883  dom cdm 5524   “ cima 5527  Fun wfun 6329  '''cafv 44063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343  df-aiota 44030  df-dfat 44065  df-afv 44066 This theorem is referenced by: (None)
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