Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfaimafn2 Structured version   Visualization version   GIF version

Theorem dfaimafn2 41721
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6396. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfaimafn2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfaimafn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfaimafn 41720 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
2 iunab 4706 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦}
31, 2syl6eqr 2800 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
4 df-sn 4310 . . . . 5 {(𝐹'''𝑥)} = {𝑦𝑦 = (𝐹'''𝑥)}
5 eqcom 2755 . . . . . 6 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
65abbii 2865 . . . . 5 {𝑦𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
74, 6eqtri 2770 . . . 4 {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
87a1i 11 . . 3 (𝑥𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
98iuneq2i 4679 . 2 𝑥𝐴 {(𝐹'''𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
103, 9syl6eqr 2800 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  {cab 2734  wrex 3039  wss 3703  {csn 4309   ciun 4660  dom cdm 5254  cima 5257  Fun wfun 6031  '''cafv 41669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045  df-dfat 41671  df-afv 41672
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator