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Theorem dfimafn2 5250
 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 5249 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
2 iunab 3730 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
31, 2syl6eqr 2106 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦})
4 df-sn 3408 . . . . 5 {(𝐹𝑥)} = {𝑦𝑦 = (𝐹𝑥)}
5 eqcom 2058 . . . . . 6 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
65abbii 2169 . . . . 5 {𝑦𝑦 = (𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
74, 6eqtri 2076 . . . 4 {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦}
87a1i 9 . . 3 (𝑥𝐴 → {(𝐹𝑥)} = {𝑦 ∣ (𝐹𝑥) = 𝑦})
98iuneq2i 3702 . 2 𝑥𝐴 {(𝐹𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹𝑥) = 𝑦}
103, 9syl6eqr 2106 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹𝑥)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  {cab 2042  ∃wrex 2324   ⊆ wss 2944  {csn 3402  ∪ ciun 3684  dom cdm 4372   “ cima 4375  Fun wfun 4923  ‘cfv 4929 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937 This theorem is referenced by: (None)
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