Theorem xpsff1o 12932

Description: The function appearing in xpsval 12935 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 12929	. . . . . 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 2580	. . . 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 6254	. . . 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 6228	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
8	7	fveq2d 5558	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
9		df-ov 5921	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
10		xp1st 6218	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
11		xp2nd 6219	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
12	4	xpsfval 12931	. . . . . . . . 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 2240	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
15	8, 14	eqtrd 2226	. . . . . 6 |- ( z e. ( A X. B ) -> ( F ` z ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
16		1st2nd2 6228	. . . . . . . 8 |- ( w e. ( A X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
17	16	fveq2d 5558	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` w ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
18		df-ov 5921	. . . . . . . 8 |- ( ( 1st ` w ) F ( 2nd ` w ) ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
19		xp1st 6218	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 1st ` w ) e. A )
20		xp2nd 6219	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 2nd ` w ) e. B )
21	4	xpsfval 12931	. . . . . . . . 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 2240	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
24	17, 23	eqtrd 2226	. . . . . 6 |- ( w e. ( A X. B ) -> ( F ` w ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
25	15, 24	eqeqan12d 2209	. . . . 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 5553	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) )
27		1stexg 6220	. . . . . . . . . 10 |- ( z e. _V -> ( 1st ` z ) e. _V )
28	27	elv 2764	. . . . . . . . 9 |- ( 1st ` z ) e. _V
29		fvpr0o 12924	. . . . . . . . 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 6220	. . . . . . . . . 10 |- ( w e. _V -> ( 1st ` w ) e. _V )
32	31	elv 2764	. . . . . . . . 9 |- ( 1st ` w ) e. _V
33		fvpr0o 12924	. . . . . . . . 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 2249	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 1st ` z ) = ( 1st ` w ) )
36		fveq1 5553	. . . . . . . 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 6221	. . . . . . . . . 10 |- ( z e. _V -> ( 2nd ` z ) e. _V )
38	37	elv 2764	. . . . . . . . 9 |- ( 2nd ` z ) e. _V
39		fvpr1o 12925	. . . . . . . . 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 6221	. . . . . . . . . 10 |- ( w e. _V -> ( 2nd ` w ) e. _V )
42	41	elv 2764	. . . . . . . . 9 |- ( 2nd ` w ) e. _V
43		fvpr1o 12925	. . . . . . . . 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 2249	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 2nd ` z ) = ( 2nd ` w ) )
46	35, 45	opeq12d 3812	. . . . . 6 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. )
47	7, 16	eqeqan12d 2209	. . . . . 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 2580	. . 3 |- A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w )
51		dff13 5811	. . 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 944	. 2 |- F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B )
53		xpsfrnel 12927	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( z Fn 2o /\ ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) )
54	53	simp2bi 1015	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` (/) ) e. A )
55	53	simp3bi 1016	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` 1o ) e. B )
56	4	xpsfval 12931	. . . . . . 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 6758	. . . . . . 7 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z Fn 2o )
59		xpsfeq 12928	. . . . . . 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 2227	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z = ( ( z ` (/) ) F ( z ` 1o ) ) )
62		rspceov 5960	. . . . 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 1249	. . . 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 2547	. . 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 6065	. . 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 944	. 2 |- F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B )
67		df-f1o 5261	. 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 944	1 |- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   (/)c0 3446   ifcif 3557   {cpr 3619   <.cop 3621   X. cxp 4657   Fn wfn 5249   -->wf 5250   -1-1->wf1 5251   -onto->wfo 5252   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   1stc1st 6191   2ndc2nd 6192   1oc1o 6462   2oc2o 6463   X_cixp 6752

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-1o 6469  df-2o 6470  df-er 6587  df-ixp 6753  df-en 6795  df-fin 6797

This theorem is referenced by:  xpsfrn  12933  xpsff1o2  12934