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Theorem dfimafn2 5536
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 5535 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
2 iunab 3912 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
31, 2eqtr4di 2217 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦})
4 df-sn 3582 . . . . 5 {(𝐹𝑥)} = {𝑦𝑦 = (𝐹𝑥)}
5 eqcom 2167 . . . . . 6 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
65abbii 2282 . . . . 5 {𝑦𝑦 = (𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
74, 6eqtri 2186 . . . 4 {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
87a1i 9 . . 3 (𝑥𝐴 → {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦})
98iuneq2i 3884 . 2 𝑥𝐴 {(𝐹𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦}
103, 9eqtr4di 2217 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1343  wcel 2136  {cab 2151  wrex 2445  wss 3116  {csn 3576   ciun 3866  dom cdm 4604  cima 4607  Fun wfun 5182  cfv 5188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196
This theorem is referenced by: (None)
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