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Theorem dffin1-5 10421
Description: Compact quantifier-free version of the standard definition df-fin 8976. (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 9031 . . . 4 (𝑥𝑦𝑦𝑥)
21rexbii 3091 . . 3 (∃𝑦 ∈ ω 𝑥𝑦 ↔ ∃𝑦 ∈ ω 𝑦𝑥)
32abbii 2798 . 2 {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
4 df-fin 8976 . 2 Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
5 dfima2 6070 . 2 ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
63, 4, 53eqtr4i 2766 1 Fin = ( ≈ “ ω)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  {cab 2705  wrex 3067   class class class wbr 5152  cima 5685  ωcom 7878  cen 8969  Fincfn 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-er 8733  df-en 8973  df-fin 8976
This theorem is referenced by: (None)
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