Theorem xpsff1o 13670

Description: The function appearing in xpsval 14201 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 13667	. . . . . 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 2636	. . . 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 6437	. . . 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 6409	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
8	7	fveq2d 5699	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
9		df-ov 6088	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
10		xp1st 6399	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
11		xp2nd 6400	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
12	4	xpsfval 13669	. . . . . . . . 9 |- ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
13	10, 11, 12	syl2anc 415	. . . . . . . 8 |- ( z e. ( A X. B ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
14	9, 13	eqtr3id 2285	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
15	8, 14	eqtrd 2271	. . . . . 6 |- ( z e. ( A X. B ) -> ( F ` z ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
16		1st2nd2 6409	. . . . . . . 8 |- ( w e. ( A X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
17	16	fveq2d 5699	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` w ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
18		df-ov 6088	. . . . . . . 8 |- ( ( 1st ` w ) F ( 2nd ` w ) ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
19		xp1st 6399	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 1st ` w ) e. A )
20		xp2nd 6400	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 2nd ` w ) e. B )
21	4	xpsfval 13669	. . . . . . . . 9 |- ( ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. B ) -> ( ( 1st ` w ) F ( 2nd ` w ) ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
22	19, 20, 21	syl2anc 415	. . . . . . . 8 |- ( w e. ( A X. B ) -> ( ( 1st ` w ) F ( 2nd ` w ) ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
23	18, 22	eqtr3id 2285	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
24	17, 23	eqtrd 2271	. . . . . 6 |- ( w e. ( A X. B ) -> ( F ` w ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
25	15, 24	eqeqan12d 2254	. . . . 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 5694	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) )
27		1stexg 6401	. . . . . . . . . 10 |- ( z e. _V -> ( 1st ` z ) e. _V )
28	27	elv 2825	. . . . . . . . 9 |- ( 1st ` z ) e. _V
29		fvpr0o 13662	. . . . . . . . 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 6401	. . . . . . . . . 10 |- ( w e. _V -> ( 1st ` w ) e. _V )
32	31	elv 2825	. . . . . . . . 9 |- ( 1st ` w ) e. _V
33		fvpr0o 13662	. . . . . . . . 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 2294	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 1st ` z ) = ( 1st ` w ) )
36		fveq1 5694	. . . . . . . 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 6402	. . . . . . . . . 10 |- ( z e. _V -> ( 2nd ` z ) e. _V )
38	37	elv 2825	. . . . . . . . 9 |- ( 2nd ` z ) e. _V
39		fvpr1o 13663	. . . . . . . . 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 6402	. . . . . . . . . 10 |- ( w e. _V -> ( 2nd ` w ) e. _V )
42	41	elv 2825	. . . . . . . . 9 |- ( 2nd ` w ) e. _V
43		fvpr1o 13663	. . . . . . . . 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 2294	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 2nd ` z ) = ( 2nd ` w ) )
46	35, 45	opeq12d 3912	. . . . . 6 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. )
47	7, 16	eqeqan12d 2254	. . . . . 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 2636	. . 3 |- A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w )
51		dff13 5974	. . 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 955	. 2 |- F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B )
53		xpsfrnel 13665	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( z Fn 2o /\ ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) )
54	53	simp2bi 1044	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` (/) ) e. A )
55	53	simp3bi 1045	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` 1o ) e. B )
56	4	xpsfval 13669	. . . . . . 7 |- ( ( ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) -> ( ( z ` (/) ) F ( z ` 1o ) ) = { <. (/) , ( z ` (/) ) >. , <. 1o , ( z ` 1o ) >. } )
57	54, 55, 56	syl2anc 415	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( ( z ` (/) ) F ( z ` 1o ) ) = { <. (/) , ( z ` (/) ) >. , <. 1o , ( z ` 1o ) >. } )
58		ixpfn 6986	. . . . . . 7 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z Fn 2o )
59		xpsfeq 13666	. . . . . . 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 2272	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z = ( ( z ` (/) ) F ( z ` 1o ) ) )
62		rspceov 6128	. . . . 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 1278	. . . 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 2603	. . 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 6236	. . 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 955	. 2 |- F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B )
67		df-f1o 5384	. 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 955	1 |- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   _Vcvv 2821   (/)c0 3520   ifcif 3638   {cpr 3710   <.cop 3712   X. cxp 4772   Fn wfn 5372   -->wf 5373   -1-1->wf1 5374   -onto->wfo 5375   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   1oc1o 6680   2oc2o 6681   X_cixp 6980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-er 6807  df-ixp 6981  df-en 7023  df-fin 7025

This theorem is used by:  xpsfrn  13671  xpsff1o2  13672