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Theorem dfimafn2 5148
 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 𝐹 A ⊆ dom 𝐹) → (𝐹A) = x A {(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 5147 . . 3 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = {yx A (𝐹x) = y})
2 iunab 3677 . . 3 x A {y ∣ (𝐹x) = y} = {yx A (𝐹x) = y}
31, 2syl6eqr 2072 . 2 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = x A {y ∣ (𝐹x) = y})
4 df-sn 3356 . . . . 5 {(𝐹x)} = {yy = (𝐹x)}
5 eqcom 2024 . . . . . 6 (y = (𝐹x) ↔ (𝐹x) = y)
65abbii 2135 . . . . 5 {yy = (𝐹x)} = {y ∣ (𝐹x) = y}
74, 6eqtri 2042 . . . 4 {(𝐹x)} = {y ∣ (𝐹x) = y}
87a1i 9 . . 3 (x A → {(𝐹x)} = {y ∣ (𝐹x) = y})
98iuneq2i 3649 . 2 x A {(𝐹x)} = x A {y ∣ (𝐹x) = y}
103, 9syl6eqr 2072 1 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = x A {(𝐹x)})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  {cab 2008  ∃wrex 2285   ⊆ wss 2894  {csn 3350  ∪ ciun 3631  dom cdm 4272   “ cima 4275  Fun wfun 4823  ‘cfv 4829 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837 This theorem is referenced by: (None)
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