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Theorem dfimafn2 5166
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 5165 . . 3 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = {yx A (𝐹x) = y})
2 iunab 3694 . . 3 x A {y ∣ (𝐹x) = y} = {yx A (𝐹x) = y}
31, 2syl6eqr 2087 . 2 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = x A {y ∣ (𝐹x) = y})
4 df-sn 3373 . . . . 5 {(𝐹x)} = {yy = (𝐹x)}
5 eqcom 2039 . . . . . 6 (y = (𝐹x) ↔ (𝐹x) = y)
65abbii 2150 . . . . 5 {yy = (𝐹x)} = {y ∣ (𝐹x) = y}
74, 6eqtri 2057 . . . 4 {(𝐹x)} = {y ∣ (𝐹x) = y}
87a1i 9 . . 3 (x A → {(𝐹x)} = {y ∣ (𝐹x) = y})
98iuneq2i 3666 . 2 x A {(𝐹x)} = x A {y ∣ (𝐹x) = y}
103, 9syl6eqr 2087 1 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = x A {(𝐹x)})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  wss 2911  {csn 3367   ciun 3648  dom cdm 4288  cima 4291  Fun wfun 4839  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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