Theorem tfinex 4486

Description: The finite T operator is always a set. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
tfinex	|- Tfin A e. _V

Proof of Theorem tfinex

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tfin 4444	. 2 |- Tfin A = if(A = (/), (/), (iotax(x e. Nn /\ E.y e. A ~P1 y e. x)))
2		0ex 4111	. . 3 |- (/) e. _V
3		iotaex 4357	. . 3 |- (iotax(x e. Nn /\ E.y e. A ~P1 y e. x)) e. _V
4	2, 3	ifex 3721	. 2 |- if(A = (/), (/), (iotax(x e. Nn /\ E.y e. A ~P1 y e. x))) e. _V
5	1, 4	eqeltri 2423	1 |- Tfin A e. _V

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616  _Vcvv 2860  (/)c0 3551  ifcif 3663  ~P₁ cpw1 4136  iotacio 4338   Nn cnnc 4374   T_fin ctfin 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-if 3664  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340  df-tfin 4444

This theorem is referenced by:  tfinltfinlem1  4501  tfinltfin  4502  eventfin  4518  oddtfin  4519  sfintfinlem1  4532  tfinnnlem1  4534  tfinnn  4535  vfinspsslem1  4551