Theorem dffin1-5 10296

Description: Compact quantifier-free version of the standard definition df-fin 8885. (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.

Step	Hyp	Ref	Expression
1		ensymb 8937	. . . 4 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑦 ≈ 𝑥)
2	1	rexbii 3081	. . 3 ⊢ (∃𝑦 ∈ ω 𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥)
3	2	abbii 2801	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
4		df-fin 8885	. 2 ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
5		dfima2 6019	. 2 ⊢ ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
6	3, 4, 5	3eqtr4i 2767	1 ⊢ Fin = ( ≈ “ ω)

Syntax hints:   = wceq 1541  {cab 2712  ∃wrex 3058   class class class wbr 5096   “ cima 5625  ωcom 7806   ≈ cen 8878  Fincfn 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-er 8633  df-en 8882  df-fin 8885

This theorem is referenced by: (None)