Theorem xpsff1o 13181

Description: The function appearing in xpsval 13184 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 13178	. . . . . 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 2592	. . . 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 6287	. . . 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 6261	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
8	7	fveq2d 5580	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
9		df-ov 5947	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
10		xp1st 6251	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
11		xp2nd 6252	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
12	4	xpsfval 13180	. . . . . . . . 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 2252	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
15	8, 14	eqtrd 2238	. . . . . 6 |- ( z e. ( A X. B ) -> ( F ` z ) = { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } )
16		1st2nd2 6261	. . . . . . . 8 |- ( w e. ( A X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
17	16	fveq2d 5580	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` w ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
18		df-ov 5947	. . . . . . . 8 |- ( ( 1st ` w ) F ( 2nd ` w ) ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
19		xp1st 6251	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 1st ` w ) e. A )
20		xp2nd 6252	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 2nd ` w ) e. B )
21	4	xpsfval 13180	. . . . . . . . 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 2252	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
24	17, 23	eqtrd 2238	. . . . . 6 |- ( w e. ( A X. B ) -> ( F ` w ) = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } )
25	15, 24	eqeqan12d 2221	. . . . 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 5575	. . . . . . . 8 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } ` (/) ) = ( { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } ` (/) ) )
27		1stexg 6253	. . . . . . . . . 10 |- ( z e. _V -> ( 1st ` z ) e. _V )
28	27	elv 2776	. . . . . . . . 9 |- ( 1st ` z ) e. _V
29		fvpr0o 13173	. . . . . . . . 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 6253	. . . . . . . . . 10 |- ( w e. _V -> ( 1st ` w ) e. _V )
32	31	elv 2776	. . . . . . . . 9 |- ( 1st ` w ) e. _V
33		fvpr0o 13173	. . . . . . . . 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 2261	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 1st ` z ) = ( 1st ` w ) )
36		fveq1 5575	. . . . . . . 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 6254	. . . . . . . . . 10 |- ( z e. _V -> ( 2nd ` z ) e. _V )
38	37	elv 2776	. . . . . . . . 9 |- ( 2nd ` z ) e. _V
39		fvpr1o 13174	. . . . . . . . 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 6254	. . . . . . . . . 10 |- ( w e. _V -> ( 2nd ` w ) e. _V )
42	41	elv 2776	. . . . . . . . 9 |- ( 2nd ` w ) e. _V
43		fvpr1o 13174	. . . . . . . . 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 2261	. . . . . . 7 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> ( 2nd ` z ) = ( 2nd ` w ) )
46	35, 45	opeq12d 3827	. . . . . 6 |- ( { <. (/) , ( 1st ` z ) >. , <. 1o , ( 2nd ` z ) >. } = { <. (/) , ( 1st ` w ) >. , <. 1o , ( 2nd ` w ) >. } -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. )
47	7, 16	eqeqan12d 2221	. . . . . 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 2592	. . 3 |- A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w )
51		dff13 5837	. . 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 13176	. . . . . 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 13180	. . . . . . 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 6791	. . . . . . 7 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z Fn 2o )
59		xpsfeq 13177	. . . . . . 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 2239	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z = ( ( z ` (/) ) F ( z ` 1o ) ) )
62		rspceov 5987	. . . . 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 2559	. . 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 6093	. . 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 5278	. 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 2176   A.wral 2484   E.wrex 2485   _Vcvv 2772   (/)c0 3460   ifcif 3571   {cpr 3634   <.cop 3636   X. cxp 4673   Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   1stc1st 6224   2ndc2nd 6225   1oc1o 6495   2oc2o 6496   X_cixp 6785

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-1o 6502  df-2o 6503  df-er 6620  df-ixp 6786  df-en 6828  df-fin 6830

This theorem is referenced by:  xpsfrn  13182  xpsff1o2  13183