Theorem xpsff1o 13512

Description: The function appearing in xpsval 13515 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = { (/) , 1o }. (Contributed by Mario Carneiro, 15-Aug-2015.)

Hypothesis

Ref	Expression
xpsff1o.f	|- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )

Assertion

Ref	Expression
xpsff1o	|- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )

Distinct variable groups:   A, k, x, y   B, k, x, y

Allowed substitution hints:   F( x, y, k)

Proof of Theorem xpsff1o

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

Step	Hyp	Ref	Expression
1		xpsfrnel2 13509	. . . . . 6 |- ( { <. (/) , x >. , <. 1o , y >. } e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( x e. A /\ y e. B ) )
2	1	biimpri 133	. . . . 5 |- ( ( x e. A /\ y e. B ) -> { <. (/) , x >. , <. 1o , y >. } e. X_ k e. 2o if ( k = (/) , A , B ) )
3	2	rgen2 2619	. . . 4 |- A. x e. A A. y e. B { <. (/) , x >. , <. 1o , y >. } e. X_ k e. 2o if ( k = (/) , A , B )
4		xpsff1o.f	. . . . 5 |- F = ( x e. A , y e. B |-> { <. (/) , x >. , <. 1o , y >. } )
5	4	fmpo 6375	. . . 4 |- ( A. x e. A A. y e. B { <. (/) , x >. , <. 1o , y >. } e. X_ k e. 2o if ( k = (/) , A , B ) <-> F : ( A X. B ) --> X_ k e. 2o if ( k = (/) , A , B ) )
6	3, 5	mpbi 145	. . 3 |- F : ( A X. B ) --> X_ k e. 2o if ( k = (/) , A , B )
7		1st2nd2 6347	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
8	7	fveq2d 5652	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
9		df-ov 6031	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
10		xp1st 6337	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
11		xp2nd 6338	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
12	4	xpsfval 13511	. . . . . . . . 9 |- ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
13	10, 11, 12	syl2anc 411	. . . . . . . 8 |- ( z e. ( A X. B ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
14	9, 13	eqtr3id 2278	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
15	8, 14	eqtrd 2264	. . . . . 6 |- ( z e. ( A X. B ) -> ( F ` z ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
16		1st2nd2 6347	. . . . . . . 8 |- ( w e. ( A X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
17	16	fveq2d 5652	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` w ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
18		df-ov 6031	. . . . . . . 8 |- ( ( 1st ` w ) F ( 2nd ` w ) ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
19		xp1st 6337	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 1st ` w ) e. A )
20		xp2nd 6338	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 2nd ` w ) e. B )
21	4	xpsfval 13511	. . . . . . . . 9 |- ( ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. B ) -> ( ( 1st ` w ) F ( 2nd ` w ) ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
22	19, 20, 21	syl2anc 411	. . . . . . . 8 |- ( w e. ( A X. B ) -> ( ( 1st ` w ) F ( 2nd ` w ) ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
23	18, 22	eqtr3id 2278	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
24	17, 23	eqtrd 2264	. . . . . 6 |- ( w e. ( A X. B ) -> ( F ` w ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
25	15, 24	eqeqan12d 2247	. . . . 5 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( ( F ` z ) = ( F ` w ) <-> { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ) )
26		fveq1 5647	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) )
27		1stexg 6339	. . . . . . . . . 10 |- ( z e. _V -> ( 1st ` z ) e. _V )
28	27	elv 2807	. . . . . . . . 9 |- ( 1st ` z ) e. _V
29		fvpr0o 13504	. . . . . . . . 9 |- ( ( 1st ` z ) e. _V -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( 1st ` z ) )
30	28, 29	ax-mp 5	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( 1st ` z )
31		1stexg 6339	. . . . . . . . . 10 |- ( w e. _V -> ( 1st ` w ) e. _V )
32	31	elv 2807	. . . . . . . . 9 |- ( 1st ` w ) e. _V
33		fvpr0o 13504	. . . . . . . . 9 |- ( ( 1st ` w ) e. _V -> ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) = ( 1st ` w ) )
34	32, 33	ax-mp 5	. . . . . . . 8 |- ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) = ( 1st ` w )
35	26, 30, 34	3eqtr3g 2287	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 1st ` z ) = ( 1st ` w ) )
36		fveq1 5647	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` 1o ) = ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` 1o ) )
37		2ndexg 6340	. . . . . . . . . 10 |- ( z e. _V -> ( 2nd ` z ) e. _V )
38	37	elv 2807	. . . . . . . . 9 |- ( 2nd ` z ) e. _V
39		fvpr1o 13505	. . . . . . . . 9 |- ( ( 2nd ` z ) e. _V -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` 1o ) = ( 2nd ` z ) )
40	38, 39	ax-mp 5	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` 1o ) = ( 2nd ` z )
41		2ndexg 6340	. . . . . . . . . 10 |- ( w e. _V -> ( 2nd ` w ) e. _V )
42	41	elv 2807	. . . . . . . . 9 |- ( 2nd ` w ) e. _V
43		fvpr1o 13505	. . . . . . . . 9 |- ( ( 2nd ` w ) e. _V -> ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` 1o ) = ( 2nd ` w ) )
44	42, 43	ax-mp 5	. . . . . . . 8 |- ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` 1o ) = ( 2nd ` w )
45	36, 40, 44	3eqtr3g 2287	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 2nd ` z ) = ( 2nd ` w ) )
46	35, 45	opeq12d 3875	. . . . . 6 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. )
47	7, 16	eqeqan12d 2247	. . . . . 6 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( z = w <-> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
48	46, 47	imbitrrid 156	. . . . 5 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> z = w ) )
49	25, 48	sylbid 150	. . . 4 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( ( F ` z ) = ( F ` w ) -> z = w ) )
50	49	rgen2 2619	. . 3 |- A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w )
51		dff13 5919	. . 3 |- ( F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B ) <-> ( F : ( A X. B ) --> X_ k e. 2o if ( k = (/) , A , B ) /\ A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w ) ) )
52	6, 50, 51	mpbir2an 951	. 2 |- F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B )
53		xpsfrnel 13507	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( z Fn 2o /\ ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) )
54	53	simp2bi 1040	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` (/) ) e. A )
55	53	simp3bi 1041	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` 1o ) e. B )
56	4	xpsfval 13511	. . . . . . 7 |- ( ( ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) -> ( ( z ` (/) ) F ( z ` 1o ) ) = { <. (/) , ( z ` (/) ) >. , <. 1o , ( z ` 1o ) >. } )
57	54, 55, 56	syl2anc 411	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( ( z ` (/) ) F ( z ` 1o ) ) = { <. (/) , ( z ` (/) ) >. , <. 1o , ( z ` 1o ) >. } )
58		ixpfn 6916	. . . . . . 7 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z Fn 2o )
59		xpsfeq 13508	. . . . . . 7 |- ( z Fn 2o -> { <. (/) , ( z ` (/) ) >. , <. 1o , ( z ` 1o ) >. } = z )
60	58, 59	syl 14	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> { <. (/) , ( z ` (/) ) >. , <. 1o , ( z ` 1o ) >. } = z )
61	57, 60	eqtr2d 2265	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z = ( ( z ` (/) ) F ( z ` 1o ) ) )
62		rspceov 6071	. . . . 5 |- ( ( ( z ` (/) ) e. A /\ ( z ` 1o ) e. B /\ z = ( ( z ` (/) ) F ( z ` 1o ) ) ) -> E. a e. A E. b e. B z = ( a F b ) )
63	54, 55, 61, 62	syl3anc 1274	. . . 4 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> E. a e. A E. b e. B z = ( a F b ) )
64	63	rgen 2586	. . 3 |- A. z e. X_ k e. 2o if ( k = (/) , A , B ) E. a e. A E. b e. B z = ( a F b )
65		foov 6179	. . 3 |- ( F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B ) <-> ( F : ( A X. B ) --> X_ k e. 2o if ( k = (/) , A , B ) /\ A. z e. X_ k e. 2o if ( k = (/) , A , B ) E. a e. A E. b e. B z = ( a F b ) ) )
66	6, 64, 65	mpbir2an 951	. 2 |- F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B )
67		df-f1o 5340	. 2 |- ( F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B ) <-> ( F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B ) /\ F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B ) ) )
68	52, 66, 67	mpbir2an 951	1 |- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   _Vcvv 2803   (/)c0 3496   ifcif 3607   {cpr 3674   <.cop 3676   X. cxp 4729   Fn wfn 5328   -->wf 5329   -1-1->wf1 5330   -onto->wfo 5331   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311   1oc1o 6618   2oc2o 6619   X_cixp 6910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-er 6745  df-ixp 6911  df-en 6953  df-fin 6955

This theorem is referenced by:  xpsfrn  13513  xpsff1o2  13514