Theorem dffin1-5 10457

Description: Compact quantifier-free version of the standard definition df-fin 9007. (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 9062	. . . 4 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑦 ≈ 𝑥)
2	1	rexbii 3100	. . 3 ⊢ (∃𝑦 ∈ ω 𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥)
3	2	abbii 2812	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
4		df-fin 9007	. 2 ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
5		dfima2 6091	. 2 ⊢ ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
6	3, 4, 5	3eqtr4i 2778	1 ⊢ Fin = ( ≈ “ ω)

Syntax hints:   = wceq 1537  {cab 2717  ∃wrex 3076   class class class wbr 5166   “ cima 5703  ωcom 7903   ≈ cen 9000  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-er 8763  df-en 9004  df-fin 9007

This theorem is referenced by: (None)