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Theorem dffin1-5 9841
 Description: Compact quantifier-free version of the standard definition df-fin 8532. (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 8576 . . . 4 (𝑥𝑦𝑦𝑥)
21rexbii 3176 . . 3 (∃𝑦 ∈ ω 𝑥𝑦 ↔ ∃𝑦 ∈ ω 𝑦𝑥)
32abbii 2824 . 2 {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
4 df-fin 8532 . 2 Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
5 dfima2 5904 . 2 ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦𝑥}
63, 4, 53eqtr4i 2792 1 Fin = ( ≈ “ ω)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1539  {cab 2736  ∃wrex 3072   class class class wbr 5033   “ cima 5528  ωcom 7580   ≈ cen 8525  Fincfn 8528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-er 8300  df-en 8529  df-fin 8532 This theorem is referenced by: (None)
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