Theorem tcex 6157

Description: The cardinal T operation always yields a set. (Contributed by SF, 2-Mar-2015.)

Assertion

Ref	Expression
tcex	|- Tc A e. _V

Proof of Theorem tcex

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

Step	Hyp	Ref	Expression
1		df-tc 6103	. 2 |- Tc A = (iotax(x e. NC /\ E.y e. A x = Nc ~P1 y))
2		iotaex 4356	. 2 |- (iotax(x e. NC /\ E.y e. A x = Nc ~P1 y)) e. _V
3	1, 2	eqeltri 2423	1 |- Tc A e. _V

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859  ~P₁ cpw1 4135  iotacio 4337   NC cncs 6088   Nc cnc 6091   T_c ctc 6093

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 4078

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339  df-tc 6103

This theorem is referenced by:  fntcfn  6245  brtcfn  6246  nmembers1lem1  6268  nchoicelem11  6299  nchoicelem16  6304