Theorem dffin1-5 10437

Description: Compact quantifier-free version of the standard definition df-fin 8955. (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 9007	. . . 4 ⊢ (𝑥 ≈ 𝑦 ↔ 𝑦 ≈ 𝑥)
2	1	rexbii 3109	. . 3 ⊢ (∃𝑦 ∈ ω 𝑥 ≈ 𝑦 ↔ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥)
3	2	abbii 2827	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
4		df-fin 8955	. 2 ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦}
5		dfima2 6052	. 2 ⊢ ( ≈ “ ω) = {𝑥 ∣ ∃𝑦 ∈ ω 𝑦 ≈ 𝑥}
6	3, 4, 5	3eqtr4i 2793	1 ⊢ Fin = ( ≈ “ ω)

This proof depends on syntax axioms:   = wceq 1570  {cab 2738  ∃wrex 3086   class class class wbr 5102   “ cima 5650  ωcom 7860   ≈ cen 8948  Fincfn 8951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-er 8695  df-en 8952  df-fin 8955

This theorem is used by: (None)