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Theorem dffin1-5 10428
Description: Compact quantifier-free version of the standard definition df-fin 8989. (Contributed by Stefan O'Rear, 6-Jan-2015.)
Assertion
Ref Expression
dffin1-5 Fin = ( ≈ “ ω)

Proof of Theorem dffin1-5
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ensymb 9042 . . . 4 (𝑥𝑦𝑦𝑥)
21rexbii 3094 . . 3 (∃𝑦 ∈ ω 𝑥𝑦 ↔ ∃𝑦 ∈ ω 𝑦𝑥)
32abbii 2809 . 2 {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
4 df-fin 8989 . 2 Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
5 dfima2 6080 . 2 ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
63, 4, 53eqtr4i 2775 1 Fin = ( ≈ “ ω)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2714  wrex 3070   class class class wbr 5143  cima 5688  ωcom 7887  cen 8982  Fincfn 8985
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-er 8745  df-en 8986  df-fin 8989
This theorem is referenced by: (None)
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