Theorem xpsff1o 13296

Description: The function appearing in xpsval 13299 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 13293	. . . . . 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 2594	. . . 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 6310	. . . 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 6284	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
8	7	fveq2d 5603	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
9		df-ov 5970	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
10		xp1st 6274	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
11		xp2nd 6275	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
12	4	xpsfval 13295	. . . . . . . . 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 2254	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
15	8, 14	eqtrd 2240	. . . . . 6 |- ( z e. ( A X. B ) -> ( F ` z ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
16		1st2nd2 6284	. . . . . . . 8 |- ( w e. ( A X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
17	16	fveq2d 5603	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` w ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
18		df-ov 5970	. . . . . . . 8 |- ( ( 1st ` w ) F ( 2nd ` w ) ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
19		xp1st 6274	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 1st ` w ) e. A )
20		xp2nd 6275	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 2nd ` w ) e. B )
21	4	xpsfval 13295	. . . . . . . . 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 2254	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
24	17, 23	eqtrd 2240	. . . . . 6 |- ( w e. ( A X. B ) -> ( F ` w ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
25	15, 24	eqeqan12d 2223	. . . . 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 5598	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) )
27		1stexg 6276	. . . . . . . . . 10 |- ( z e. _V -> ( 1st ` z ) e. _V )
28	27	elv 2780	. . . . . . . . 9 |- ( 1st ` z ) e. _V
29		fvpr0o 13288	. . . . . . . . 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 6276	. . . . . . . . . 10 |- ( w e. _V -> ( 1st ` w ) e. _V )
32	31	elv 2780	. . . . . . . . 9 |- ( 1st ` w ) e. _V
33		fvpr0o 13288	. . . . . . . . 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 2263	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 1st ` z ) = ( 1st ` w ) )
36		fveq1 5598	. . . . . . . 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 6277	. . . . . . . . . 10 |- ( z e. _V -> ( 2nd ` z ) e. _V )
38	37	elv 2780	. . . . . . . . 9 |- ( 2nd ` z ) e. _V
39		fvpr1o 13289	. . . . . . . . 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 6277	. . . . . . . . . 10 |- ( w e. _V -> ( 2nd ` w ) e. _V )
42	41	elv 2780	. . . . . . . . 9 |- ( 2nd ` w ) e. _V
43		fvpr1o 13289	. . . . . . . . 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 2263	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 2nd ` z ) = ( 2nd ` w ) )
46	35, 45	opeq12d 3841	. . . . . 6 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. )
47	7, 16	eqeqan12d 2223	. . . . . 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 2594	. . 3 |- A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w )
51		dff13 5860	. . 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 945	. 2 |- F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B )
53		xpsfrnel 13291	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( z Fn 2o /\ ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) )
54	53	simp2bi 1016	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` (/) ) e. A )
55	53	simp3bi 1017	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` 1o ) e. B )
56	4	xpsfval 13295	. . . . . . 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 6814	. . . . . . 7 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z Fn 2o )
59		xpsfeq 13292	. . . . . . 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 2241	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z = ( ( z ` (/) ) F ( z ` 1o ) ) )
62		rspceov 6010	. . . . 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 1250	. . . 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 2561	. . 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 6116	. . 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 945	. 2 |- F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B )
67		df-f1o 5297	. 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 945	1 |- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   _Vcvv 2776   (/)c0 3468   ifcif 3579   {cpr 3644   <.cop 3646   X. cxp 4691   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   -onto->wfo 5288   -1-1-onto->wf1o 5289   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   1stc1st 6247   2ndc2nd 6248   1oc1o 6518   2oc2o 6519   X_cixp 6808

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-1o 6525  df-2o 6526  df-er 6643  df-ixp 6809  df-en 6851  df-fin 6853

This theorem is referenced by:  xpsfrn  13297  xpsff1o2  13298