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Theorem dfimafn2 5533
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 5532 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
2 iunab 3909 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
31, 2eqtr4di 2215 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦})
4 df-sn 3579 . . . . 5 {(𝐹𝑥)} = {𝑦𝑦 = (𝐹𝑥)}
5 eqcom 2166 . . . . . 6 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
65abbii 2280 . . . . 5 {𝑦𝑦 = (𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
74, 6eqtri 2185 . . . 4 {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
87a1i 9 . . 3 (𝑥𝐴 → {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦})
98iuneq2i 3881 . 2 𝑥𝐴 {(𝐹𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦}
103, 9eqtr4di 2215 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1342  wcel 2135  {cab 2150  wrex 2443  wss 3114  {csn 3573   ciun 3863  dom cdm 4601  cima 4604  Fun wfun 5179  cfv 5185
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-sbc 2950  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-iun 3865  df-br 3980  df-opab 4041  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-fv 5193
This theorem is referenced by: (None)
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