Theorem tfinprop 4490

Description: Properties of the finite T operator for a nonempty natural. Theorem X.1.28 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
tfinprop	|- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M))

Distinct variable group:   M,a

Proof of Theorem tfinprop

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tfin 4444	. . 3 |- Tfin M = if(M = (/), (/), (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)))
2		df-ne 2519	. . . . . 6 |- (M =/= (/) <-> -. M = (/))
3		iffalse 3670	. . . . . 6 |- (-. M = (/) -> if(M = (/), (/), (iotan(n e. Nn /\ E.a e. M ~P1 a e. n))) = (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)))
4	2, 3	sylbi 187	. . . . 5 |- (M =/= (/) -> if(M = (/), (/), (iotan(n e. Nn /\ E.a e. M ~P1 a e. n))) = (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)))
5	4	adantl 452	. . . 4 |- ((M e. Nn /\ M =/= (/)) -> if(M = (/), (/), (iotan(n e. Nn /\ E.a e. M ~P1 a e. n))) = (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)))
6		nnpw1ex 4485	. . . . 5 |- ((M e. Nn /\ M =/= (/)) -> E!n e. Nn E.a e. M ~P1 a e. n)
7		reiotacl 4365	. . . . 5 |- (E!n e. Nn E.a e. M ~P1 a e. n -> (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)) e. Nn )
8	6, 7	syl 15	. . . 4 |- ((M e. Nn /\ M =/= (/)) -> (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)) e. Nn )
9	5, 8	eqeltrd 2427	. . 3 |- ((M e. Nn /\ M =/= (/)) -> if(M = (/), (/), (iotan(n e. Nn /\ E.a e. M ~P1 a e. n))) e. Nn )
10	1, 9	syl5eqel 2437	. 2 |- ((M e. Nn /\ M =/= (/)) -> Tfin M e. Nn )
11	1, 5	syl5req 2398	. . 3 |- ((M e. Nn /\ M =/= (/)) -> (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)) = Tfin M)
12	10, 6	jca 518	. . . 4 |- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E!n e. Nn E.a e. M ~P1 a e. n))
13		eleq2 2414	. . . . . 6 |- (n = Tfin M -> (~P1 a e. n <-> ~P1 a e. Tfin M))
14	13	rexbidv 2636	. . . . 5 |- (n = Tfin M -> (E.a e. M ~P1 a e. n <-> E.a e. M ~P1 a e. Tfin M))
15	14	reiota2 4369	. . . 4 |- (( Tfin M e. Nn /\ E!n e. Nn E.a e. M ~P1 a e. n) -> (E.a e. M ~P1 a e. Tfin M <-> (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)) = Tfin M))
16	12, 15	syl 15	. . 3 |- ((M e. Nn /\ M =/= (/)) -> (E.a e. M ~P1 a e. Tfin M <-> (iotan(n e. Nn /\ E.a e. M ~P1 a e. n)) = Tfin M))
17	11, 16	mpbird 223	. 2 |- ((M e. Nn /\ M =/= (/)) -> E.a e. M ~P1 a e. Tfin M)
18	10, 17	jca 518	1 |- ((M e. Nn /\ M =/= (/)) -> ( Tfin M e. Nn /\ E.a e. M ~P1 a e. Tfin M))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  E!wreu 2617  (/)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  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-tfin 4444

This theorem is referenced by:  tfinnnul  4491  tfincl  4493  tfin11  4494  tfinpw1  4495  tfinltfinlem1  4501  tfinltfin  4502  eventfin  4518  oddtfin  4519  sfinltfin  4536  vfinncvntnn  4549