Theorem tfineq 4489

Description: Equality theorem for the finite T operator. (Contributed by SF, 24-Jan-2015.)

Assertion

Ref	Expression
tfineq	|- (A = B -> Tfin A = Tfin B)

Proof of Theorem tfineq

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

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . 3 |- (A = B -> (A = (/) <-> B = (/)))
2		rexeq 2809	. . . . 5 |- (A = B -> (E.y e. A ~P1 y e. x <-> E.y e. B ~P1 y e. x))
3	2	anbi2d 684	. . . 4 |- (A = B -> ((x e. Nn /\ E.y e. A ~P1 y e. x) <-> (x e. Nn /\ E.y e. B ~P1 y e. x)))
4	3	iotabidv 4361	. . 3 |- (A = B -> (iotax(x e. Nn /\ E.y e. A ~P1 y e. x)) = (iotax(x e. Nn /\ E.y e. B ~P1 y e. x)))
5	1, 4	ifbieq2d 3683	. 2 |- (A = B -> if(A = (/), (/), (iotax(x e. Nn /\ E.y e. A ~P1 y e. x))) = if(B = (/), (/), (iotax(x e. Nn /\ E.y e. B ~P1 y e. x))))
6		df-tfin 4444	. 2 |- Tfin A = if(A = (/), (/), (iotax(x e. Nn /\ E.y e. A ~P1 y e. x)))
7		df-tfin 4444	. 2 |- Tfin B = if(B = (/), (/), (iotax(x e. Nn /\ E.y e. B ~P1 y e. x)))
8	5, 6, 7	3eqtr4g 2410	1 |- (A = B -> Tfin A = Tfin B)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616  (/)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

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664  df-uni 3893  df-iota 4340  df-tfin 4444

This theorem is referenced by:  tfincl  4493  tfin11  4494  tfin1c  4500  tfinltfinlem1  4501  tfinltfin  4502  eventfin  4518  oddtfin  4519  sfintfin  4533  tfinnn  4535  vfinncvntnn  4549  vfinspsslem1  4551  vfinspss  4552  vfinspclt  4553