Theorem pw1fnf1o 5856

Description: Pw1Fn is a one-to-one function with domain 1_c and range ~P1_c. (Contributed by SF, 26-Feb-2015.)

Assertion

Ref	Expression
pw1fnf1o	|- Pw1Fn :1c-1-1-onto->~P1c

Proof of Theorem pw1fnf1o

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

Step	Hyp	Ref	Expression
1		fnpw1fn 5854	. 2 |- Pw1Fn Fn 1c
2		df-pw1fn 5767	. . . 4 |- Pw1Fn = (x e. 1c |-> ~P1 U.x)
3	2	rnmpt 5687	. . 3 |- ran Pw1Fn = {y | E.x e. 1c y = ~P1 U.x}
4		vex 2863	. . . . . . 7 |- y e. _V
5	4	sspw1 4336	. . . . . 6 |- (y C_ ~P1 _V <-> E.z(z C_ _V /\ y = ~P1 z))
6		df1c2 4169	. . . . . . 7 |- 1c = ~P1 _V
7	6	sseq2i 3297	. . . . . 6 |- (y C_ 1c <-> y C_ ~P1 _V)
8		ssv 3292	. . . . . . . 8 |- z C_ _V
9	8	biantrur 492	. . . . . . 7 |- (y = ~P1 z <-> (z C_ _V /\ y = ~P1 z))
10	9	exbii 1582	. . . . . 6 |- (E.z y = ~P1 z <-> E.z(z C_ _V /\ y = ~P1 z))
11	5, 7, 10	3bitr4i 268	. . . . 5 |- (y C_ 1c <-> E.z y = ~P1 z)
12	4	elpw 3729	. . . . 5 |- (y e. ~P1c <-> y C_ 1c)
13		df-rex 2621	. . . . . 6 |- (E.x e. 1c y = ~P1 U.x <-> E.x(x e. 1c /\ y = ~P1 U.x))
14		el1c 4140	. . . . . . . . 9 |- (x e. 1c <-> E.z x = {z})
15	14	anbi1i 676	. . . . . . . 8 |- ((x e. 1c /\ y = ~P1 U.x) <-> (E.z x = {z} /\ y = ~P1 U.x))
16		19.41v 1901	. . . . . . . 8 |- (E.z(x = {z} /\ y = ~P1 U.x) <-> (E.z x = {z} /\ y = ~P1 U.x))
17	15, 16	bitr4i 243	. . . . . . 7 |- ((x e. 1c /\ y = ~P1 U.x) <-> E.z(x = {z} /\ y = ~P1 U.x))
18	17	exbii 1582	. . . . . 6 |- (E.x(x e. 1c /\ y = ~P1 U.x) <-> E.xE.z(x = {z} /\ y = ~P1 U.x))
19		excom 1741	. . . . . . 7 |- (E.xE.z(x = {z} /\ y = ~P1 U.x) <-> E.zE.x(x = {z} /\ y = ~P1 U.x))
20		snex 4112	. . . . . . . . 9 |- {z} e. _V
21		unieq 3901	. . . . . . . . . . . 12 |- (x = {z} -> U.x = U.{z})
22		vex 2863	. . . . . . . . . . . . 13 |- z e. _V
23	22	unisn 3908	. . . . . . . . . . . 12 |- U.{z} = z
24	21, 23	syl6eq 2401	. . . . . . . . . . 11 |- (x = {z} -> U.x = z)
25		pw1eq 4144	. . . . . . . . . . 11 |- (U.x = z -> ~P1 U.x = ~P1 z)
26	24, 25	syl 15	. . . . . . . . . 10 |- (x = {z} -> ~P1 U.x = ~P1 z)
27	26	eqeq2d 2364	. . . . . . . . 9 |- (x = {z} -> (y = ~P1 U.x <-> y = ~P1 z))
28	20, 27	ceqsexv 2895	. . . . . . . 8 |- (E.x(x = {z} /\ y = ~P1 U.x) <-> y = ~P1 z)
29	28	exbii 1582	. . . . . . 7 |- (E.zE.x(x = {z} /\ y = ~P1 U.x) <-> E.z y = ~P1 z)
30	19, 29	bitri 240	. . . . . 6 |- (E.xE.z(x = {z} /\ y = ~P1 U.x) <-> E.z y = ~P1 z)
31	13, 18, 30	3bitri 262	. . . . 5 |- (E.x e. 1c y = ~P1 U.x <-> E.z y = ~P1 z)
32	11, 12, 31	3bitr4i 268	. . . 4 |- (y e. ~P1c <-> E.x e. 1c y = ~P1 U.x)
33	32	eqabi 2465	. . 3 |- ~P1c = {y | E.x e. 1c y = ~P1 U.x}
34	3, 33	eqtr4i 2376	. 2 |- ran Pw1Fn = ~P1c
35		el1c 4140	. . . . . 6 |- (x e. 1c <-> E.a x = {a})
36		el1c 4140	. . . . . 6 |- (y e. 1c <-> E.b y = {b})
37	35, 36	anbi12i 678	. . . . 5 |- ((x e. 1c /\ y e. 1c) <-> (E.a x = {a} /\ E.b y = {b}))
38		eeanv 1913	. . . . 5 |- (E.aE.b(x = {a} /\ y = {b}) <-> (E.a x = {a} /\ E.b y = {b}))
39	37, 38	bitr4i 243	. . . 4 |- ((x e. 1c /\ y e. 1c) <-> E.aE.b(x = {a} /\ y = {b}))
40		pw111 4171	. . . . . . . 8 |- (~P1 a = ~P1 b <-> a = b)
41	40	biimpi 186	. . . . . . 7 |- (~P1 a = ~P1 b -> a = b)
42	41	a1i 10	. . . . . 6 |- ((x = {a} /\ y = {b}) -> (~P1 a = ~P1 b -> a = b))
43		fveq2 5329	. . . . . . . 8 |- (x = {a} -> ( Pw1Fn ` x) = ( Pw1Fn ` {a}))
44		vex 2863	. . . . . . . . 9 |- a e. _V
45	44	pw1fnval 5852	. . . . . . . 8 |- ( Pw1Fn ` {a}) = ~P1 a
46	43, 45	syl6eq 2401	. . . . . . 7 |- (x = {a} -> ( Pw1Fn ` x) = ~P1 a)
47		fveq2 5329	. . . . . . . 8 |- (y = {b} -> ( Pw1Fn ` y) = ( Pw1Fn ` {b}))
48		vex 2863	. . . . . . . . 9 |- b e. _V
49	48	pw1fnval 5852	. . . . . . . 8 |- ( Pw1Fn ` {b}) = ~P1 b
50	47, 49	syl6eq 2401	. . . . . . 7 |- (y = {b} -> ( Pw1Fn ` y) = ~P1 b)
51	46, 50	eqeqan12d 2368	. . . . . 6 |- ((x = {a} /\ y = {b}) -> (( Pw1Fn ` x) = ( Pw1Fn ` y) <-> ~P1 a = ~P1 b))
52		eqeq12 2365	. . . . . . 7 |- ((x = {a} /\ y = {b}) -> (x = y <-> {a} = {b}))
53	44	sneqb 3877	. . . . . . 7 |- ({a} = {b} <-> a = b)
54	52, 53	syl6bb 252	. . . . . 6 |- ((x = {a} /\ y = {b}) -> (x = y <-> a = b))
55	42, 51, 54	3imtr4d 259	. . . . 5 |- ((x = {a} /\ y = {b}) -> (( Pw1Fn ` x) = ( Pw1Fn ` y) -> x = y))
56	55	exlimivv 1635	. . . 4 |- (E.aE.b(x = {a} /\ y = {b}) -> (( Pw1Fn ` x) = ( Pw1Fn ` y) -> x = y))
57	39, 56	sylbi 187	. . 3 |- ((x e. 1c /\ y e. 1c) -> (( Pw1Fn ` x) = ( Pw1Fn ` y) -> x = y))
58	57	rgen2a 2681	. 2 |- A.x e. 1c A.y e. 1c (( Pw1Fn ` x) = ( Pw1Fn ` y) -> x = y)
59		dff1o6 5476	. 2 |- ( Pw1Fn :1c-1-1-onto->~P1c <-> ( Pw1Fn Fn 1c /\ ran Pw1Fn = ~P1c /\ A.x e. 1c A.y e. 1c (( Pw1Fn ` x) = ( Pw1Fn ` y) -> x = y)))
60	1, 34, 58, 59	mpbir3an 1134	1 |- Pw1Fn :1c-1-1-onto->~P1c

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  E.wrex 2616  _Vcvv 2860   C_ wss 3258  ~Pcpw 3723  {csn 3738  U.cuni 3892  1_cc1c 4135  ~P₁ cpw1 4136  ran crn 4774   Fn wfn 4777  -1-1-onto->wf1o 4781  ` cfv 4782   Pw1Fn cpw1fn 5766

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-13 1712  ax-14 1714  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-3or 935  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-pss 3262  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-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-mpt 5653  df-pw1fn 5767

This theorem is referenced by:  enpw1pw  6076