Theorem xpsff1o 16228

Description: The function appearing in xpsval 16232 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 } +c { y } ) )

Assertion

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

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

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

Proof of Theorem xpsff1o

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

Step	Hyp	Ref	Expression
1		xpsfrnel2 16225	. . . . . 6 |- ( `' ( { x } +c { y } ) e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( x e. A /\ y e. B ) )
2	1	biimpri 218	. . . . 5 |- ( ( x e. A /\ y e. B ) -> `' ( { x } +c { y } ) e. X_ k e. 2o if ( k = (/) , A , B ) )
3	2	rgen2 2975	. . . 4 |- A. x e. A A. y e. B `' ( { x } +c { y } ) e. X_ k e. 2o if ( k = (/) , A , B )
4		xpsff1o.f	. . . . 5 |- F = ( x e. A , y e. B |-> `' ( { x } +c { y } ) )
5	4	fmpt2 7237	. . . 4 |- ( A. x e. A A. y e. B `' ( { x } +c { 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 220	. . 3 |- F : ( A X. B ) --> X_ k e. 2o if ( k = (/) , A , B )
7		1st2nd2 7205	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
8	7	fveq2d 6195	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` z ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) )
9		df-ov 6653	. . . . . . . 8 |- ( ( 1st ` z ) F ( 2nd ` z ) ) = ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. )
10		xp1st 7198	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
11		xp2nd 7199	. . . . . . . . 9 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
12	4	xpsfval 16227	. . . . . . . . 9 |- ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) )
13	10, 11, 12	syl2anc 693	. . . . . . . 8 |- ( z e. ( A X. B ) -> ( ( 1st ` z ) F ( 2nd ` z ) ) = `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) )
14	9, 13	syl5eqr 2670	. . . . . . 7 |- ( z e. ( A X. B ) -> ( F ` <. ( 1st ` z ) , ( 2nd ` z ) >. ) = `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) )
15	8, 14	eqtrd 2656	. . . . . 6 |- ( z e. ( A X. B ) -> ( F ` z ) = `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) )
16		1st2nd2 7205	. . . . . . . 8 |- ( w e. ( A X. B ) -> w = <. ( 1st ` w ) , ( 2nd ` w ) >. )
17	16	fveq2d 6195	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` w ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
18		df-ov 6653	. . . . . . . 8 |- ( ( 1st ` w ) F ( 2nd ` w ) ) = ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. )
19		xp1st 7198	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 1st ` w ) e. A )
20		xp2nd 7199	. . . . . . . . 9 |- ( w e. ( A X. B ) -> ( 2nd ` w ) e. B )
21	4	xpsfval 16227	. . . . . . . . 9 |- ( ( ( 1st ` w ) e. A /\ ( 2nd ` w ) e. B ) -> ( ( 1st ` w ) F ( 2nd ` w ) ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) )
22	19, 20, 21	syl2anc 693	. . . . . . . 8 |- ( w e. ( A X. B ) -> ( ( 1st ` w ) F ( 2nd ` w ) ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) )
23	18, 22	syl5eqr 2670	. . . . . . 7 |- ( w e. ( A X. B ) -> ( F ` <. ( 1st ` w ) , ( 2nd ` w ) >. ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) )
24	17, 23	eqtrd 2656	. . . . . 6 |- ( w e. ( A X. B ) -> ( F ` w ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) )
25	15, 24	eqeqan12d 2638	. . . . 5 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( ( F ` z ) = ( F ` w ) <-> `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ) )
26		fveq1 6190	. . . . . . . 8 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) -> ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) ` (/) ) = ( `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ` (/) ) )
27		fvex 6201	. . . . . . . . 9 |- ( 1st ` z ) e. _V
28		xpsc0 16220	. . . . . . . . 9 |- ( ( 1st ` z ) e. _V -> ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) ` (/) ) = ( 1st ` z ) )
29	27, 28	ax-mp 5	. . . . . . . 8 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) ` (/) ) = ( 1st ` z )
30		fvex 6201	. . . . . . . . 9 |- ( 1st ` w ) e. _V
31		xpsc0 16220	. . . . . . . . 9 |- ( ( 1st ` w ) e. _V -> ( `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ` (/) ) = ( 1st ` w ) )
32	30, 31	ax-mp 5	. . . . . . . 8 |- ( `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ` (/) ) = ( 1st ` w )
33	26, 29, 32	3eqtr3g 2679	. . . . . . 7 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) -> ( 1st ` z ) = ( 1st ` w ) )
34		fveq1 6190	. . . . . . . 8 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) -> ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) ` 1o ) = ( `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ` 1o ) )
35		fvex 6201	. . . . . . . . 9 |- ( 2nd ` z ) e. _V
36		xpsc1 16221	. . . . . . . . 9 |- ( ( 2nd ` z ) e. _V -> ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) ` 1o ) = ( 2nd ` z ) )
37	35, 36	ax-mp 5	. . . . . . . 8 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) ` 1o ) = ( 2nd ` z )
38		fvex 6201	. . . . . . . . 9 |- ( 2nd ` w ) e. _V
39		xpsc1 16221	. . . . . . . . 9 |- ( ( 2nd ` w ) e. _V -> ( `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ` 1o ) = ( 2nd ` w ) )
40	38, 39	ax-mp 5	. . . . . . . 8 |- ( `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) ` 1o ) = ( 2nd ` w )
41	34, 37, 40	3eqtr3g 2679	. . . . . . 7 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) -> ( 2nd ` z ) = ( 2nd ` w ) )
42	33, 41	opeq12d 4410	. . . . . 6 |- ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. )
43	7, 16	eqeqan12d 2638	. . . . . 6 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( z = w <-> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. ( 1st ` w ) , ( 2nd ` w ) >. ) )
44	42, 43	syl5ibr 236	. . . . 5 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( `' ( { ( 1st ` z ) } +c { ( 2nd ` z ) } ) = `' ( { ( 1st ` w ) } +c { ( 2nd ` w ) } ) -> z = w ) )
45	25, 44	sylbid 230	. . . 4 |- ( ( z e. ( A X. B ) /\ w e. ( A X. B ) ) -> ( ( F ` z ) = ( F ` w ) -> z = w ) )
46	45	rgen2 2975	. . 3 |- A. z e. ( A X. B ) A. w e. ( A X. B ) ( ( F ` z ) = ( F ` w ) -> z = w )
47		dff13 6512	. . 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 ) ) )
48	6, 46, 47	mpbir2an 955	. 2 |- F : ( A X. B ) -1-1-> X_ k e. 2o if ( k = (/) , A , B )
49		xpsfrnel 16223	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( z Fn 2o /\ ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) )
50	49	simp2bi 1077	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` (/) ) e. A )
51	49	simp3bi 1078	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( z ` 1o ) e. B )
52	4	xpsfval 16227	. . . . . . 7 |- ( ( ( z ` (/) ) e. A /\ ( z ` 1o ) e. B ) -> ( ( z ` (/) ) F ( z ` 1o ) ) = `' ( { ( z ` (/) ) } +c { ( z ` 1o ) } ) )
53	50, 51, 52	syl2anc 693	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> ( ( z ` (/) ) F ( z ` 1o ) ) = `' ( { ( z ` (/) ) } +c { ( z ` 1o ) } ) )
54		ixpfn 7914	. . . . . . 7 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z Fn 2o )
55		xpsfeq 16224	. . . . . . 7 |- ( z Fn 2o -> `' ( { ( z ` (/) ) } +c { ( z ` 1o ) } ) = z )
56	54, 55	syl 17	. . . . . 6 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> `' ( { ( z ` (/) ) } +c { ( z ` 1o ) } ) = z )
57	53, 56	eqtr2d 2657	. . . . 5 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> z = ( ( z ` (/) ) F ( z ` 1o ) ) )
58		rspceov 6692	. . . . 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 ) )
59	50, 51, 57, 58	syl3anc 1326	. . . 4 |- ( z e. X_ k e. 2o if ( k = (/) , A , B ) -> E. a e. A E. b e. B z = ( a F b ) )
60	59	rgen 2922	. . 3 |- A. z e. X_ k e. 2o if ( k = (/) , A , B ) E. a e. A E. b e. B z = ( a F b )
61		foov 6808	. . 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 ) ) )
62	6, 60, 61	mpbir2an 955	. 2 |- F : ( A X. B ) -onto-> X_ k e. 2o if ( k = (/) , A , B )
63		df-f1o 5895	. 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 ) ) )
64	48, 62, 63	mpbir2an 955	1 |- F : ( A X. B ) -1-1-onto-> X_ k e. 2o if ( k = (/) , A , B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   <.cop 4183   X. cxp 5112   `'ccnv 5113   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   1oc1o 7553   2oc2o 7554   X_cixp 7908   +c ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-cda 8990

This theorem is referenced by:  xpsfrn  16229  xpsff1o2  16231