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Theorem dfaimafn2 46172
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 6954. (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 46171 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦})
2 iunab 5053 . . 3 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 (𝐹'''𝑥) = 𝑦}
31, 2eqtr4di 2788 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
4 df-sn 4628 . . . . 5 {(𝐹'''𝑥)} = {𝑦𝑦 = (𝐹'''𝑥)}
5 eqcom 2737 . . . . . 6 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
65abbii 2800 . . . . 5 {𝑦𝑦 = (𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
74, 6eqtri 2758 . . . 4 {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
87a1i 11 . . 3 (𝑥𝐴 → {(𝐹'''𝑥)} = {𝑦 ∣ (𝐹'''𝑥) = 𝑦})
98iuneq2i 5017 . 2 𝑥𝐴 {(𝐹'''𝑥)} = 𝑥𝐴 {𝑦 ∣ (𝐹'''𝑥) = 𝑦}
103, 9eqtr4di 2788 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 {(𝐹'''𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {cab 2707  wrex 3068  wss 3947  {csn 4627   ciun 4996  dom cdm 5675  cima 5678  Fun wfun 6536  '''cafv 46123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-aiota 46091  df-dfat 46125  df-afv 46126
This theorem is referenced by: (None)
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