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Theorem dfimafn 5165
Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
dfimafn ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = {yx A (𝐹x) = y})
Distinct variable groups:   x,y,A   x,𝐹,y

Proof of Theorem dfimafn
StepHypRef Expression
1 ssel 2933 . . . . . 6 (A ⊆ dom 𝐹 → (x Ax dom 𝐹))
2 funbrfvb 5159 . . . . . . 7 ((Fun 𝐹 x dom 𝐹) → ((𝐹x) = yx𝐹y))
32ex 108 . . . . . 6 (Fun 𝐹 → (x dom 𝐹 → ((𝐹x) = yx𝐹y)))
41, 3syl9r 67 . . . . 5 (Fun 𝐹 → (A ⊆ dom 𝐹 → (x A → ((𝐹x) = yx𝐹y))))
54imp31 243 . . . 4 (((Fun 𝐹 A ⊆ dom 𝐹) x A) → ((𝐹x) = yx𝐹y))
65rexbidva 2317 . . 3 ((Fun 𝐹 A ⊆ dom 𝐹) → (x A (𝐹x) = yx A x𝐹y))
76abbidv 2152 . 2 ((Fun 𝐹 A ⊆ dom 𝐹) → {yx A (𝐹x) = y} = {yx A x𝐹y})
8 dfima2 4613 . 2 (𝐹A) = {yx A x𝐹y}
97, 8syl6reqr 2088 1 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) = {yx A (𝐹x) = y})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  wss 2911   class class class wbr 3755  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-bndl 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-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:  dfimafn2  5166  fvelimab  5172
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