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Theorem dffin1-5 9804
Description: Compact quantifier-free version of the standard definition df-fin 8507. (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 8551 . . . 4 (𝑥𝑦𝑦𝑥)
21rexbii 3252 . . 3 (∃𝑦 ∈ ω 𝑥𝑦 ↔ ∃𝑦 ∈ ω 𝑦𝑥)
32abbii 2891 . 2 {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
4 df-fin 8507 . 2 Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
5 dfima2 5930 . 2 ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
63, 4, 53eqtr4i 2859 1 Fin = ( ≈ “ ω)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  {cab 2804  wrex 3144   class class class wbr 5063  cima 5557  ωcom 7573  cen 8500  Fincfn 8503
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-er 8284  df-en 8504  df-fin 8507
This theorem is referenced by: (None)
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