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 → (∃y ∈ A ℘1y ∈ x ↔ ∃y ∈ B ℘1y ∈ x))
3	2	anbi2d 684	. . . 4 ⊢ (A = B → ((x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x) ↔ (x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x)))
4	3	iotabidv 4361	. . 3 ⊢ (A = B → (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)) = (℩x(x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x)))
5	1, 4	ifbieq2d 3683	. 2 ⊢ (A = B → if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x))) = if(B = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x))))
6		df-tfin 4444	. 2 ⊢ Tfin A = if(A = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ A ℘1y ∈ x)))
7		df-tfin 4444	. 2 ⊢ Tfin B = if(B = ∅, ∅, (℩x(x ∈ Nn ∧ ∃y ∈ B ℘1y ∈ x)))
8	5, 6, 7	3eqtr4g 2410	1 ⊢ (A = B → Tfin A = Tfin B)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∅c0 3551   ifcif 3663  ℘₁cpw1 4136  ℩cio 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