Theorem dffin1-5 10341

Description: Compact quantifier-free version of the standard definition df-fin 8922. (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 8973	. . . 4 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑦 ≈ 𝑥)
2	1	rexbii 3076	. . 3 ⊢ (∃𝑦 ∈ ω 𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥)
3	2	abbii 2796	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
4		df-fin 8922	. 2 ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
5		dfima2 6033	. 2 ⊢ ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
6	3, 4, 5	3eqtr4i 2762	1 ⊢ Fin = ( ≈ “ ω)

Syntax hints:   = wceq 1540  {cab 2707  ∃wrex 3053   class class class wbr 5107   “ cima 5641  ωcom 7842   ≈ cen 8915  Fincfn 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-er 8671  df-en 8919  df-fin 8922

This theorem is referenced by: (None)