MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffin1-5 Structured version   Visualization version   GIF version

Theorem dffin1-5 10426
Description: Compact quantifier-free version of the standard definition df-fin 8988. (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 9041 . . . 4 (𝑥𝑦𝑦𝑥)
21rexbii 3092 . . 3 (∃𝑦 ∈ ω 𝑥𝑦 ↔ ∃𝑦 ∈ ω 𝑦𝑥)
32abbii 2807 . 2 {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
4 df-fin 8988 . 2 Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
5 dfima2 6082 . 2 ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
63, 4, 53eqtr4i 2773 1 Fin = ( ≈ “ ω)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {cab 2712  wrex 3068   class class class wbr 5148  cima 5692  ωcom 7887  cen 8981  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-er 8744  df-en 8985  df-fin 8988
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator