Theorem f1o3d 6271

Description: Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)

Hypotheses

Ref	Expression
f1o3d.1	|- ( ph -> F = ( x e. A |-> C ) )
f1o3d.2	|- ( ( ph /\ x e. A ) -> C e. B )
f1o3d.3	|- ( ( ph /\ y e. B ) -> D e. A )
f1o3d.4	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )

Assertion

Ref	Expression
f1o3d	|- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Distinct variable groups:   x, y, A   x, B, y   y, C   x, D   ph, x, y

Allowed substitution hints:   C( x)   D( y)   F( x, y)

Proof of Theorem f1o3d

Step	Hyp	Ref	Expression
1		f1o3d.2	. . . . . 6 |- ( ( ph /\ x e. A ) -> C e. B )
2	1	ralrimiva 2617	. . . . 5 |- ( ph -> A. x e. A C e. B )
3		eqid 2234	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
4	3	fnmpt 5490	. . . . 5 |- ( A. x e. A C e. B -> ( x e. A |-> C ) Fn A )
5	2, 4	syl 14	. . . 4 |- ( ph -> ( x e. A |-> C ) Fn A )
6		f1o3d.1	. . . . 5 |- ( ph -> F = ( x e. A |-> C ) )
7	6	fneq1d 5451	. . . 4 |- ( ph -> ( F Fn A <-> ( x e. A |-> C ) Fn A ) )
8	5, 7	mpbird 167	. . 3 |- ( ph -> F Fn A )
9		f1o3d.3	. . . . . 6 |- ( ( ph /\ y e. B ) -> D e. A )
10	9	ralrimiva 2617	. . . . 5 |- ( ph -> A. y e. B D e. A )
11		eqid 2234	. . . . . 6 |- ( y e. B |-> D ) = ( y e. B |-> D )
12	11	fnmpt 5490	. . . . 5 |- ( A. y e. B D e. A -> ( y e. B |-> D ) Fn B )
13	10, 12	syl 14	. . . 4 |- ( ph -> ( y e. B |-> D ) Fn B )
14		eleq1a 2306	. . . . . . . . . . 11 |- ( C e. B -> ( y = C -> y e. B ) )
15	1, 14	syl 14	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( y = C -> y e. B ) )
16	15	impr 379	. . . . . . . . 9 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> y e. B )
17		f1o3d.4	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )
18	17	biimpar 297	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ y = C ) -> x = D )
19	18	exp42 371	. . . . . . . . . . 11 |- ( ph -> ( x e. A -> ( y e. B -> ( y = C -> x = D ) ) ) )
20	19	com34 83	. . . . . . . . . 10 |- ( ph -> ( x e. A -> ( y = C -> ( y e. B -> x = D ) ) ) )
21	20	imp32 257	. . . . . . . . 9 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B -> x = D ) )
22	16, 21	jcai 311	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B /\ x = D ) )
23		eleq1a 2306	. . . . . . . . . . 11 |- ( D e. A -> ( x = D -> x e. A ) )
24	9, 23	syl 14	. . . . . . . . . 10 |- ( ( ph /\ y e. B ) -> ( x = D -> x e. A ) )
25	24	impr 379	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> x e. A )
26	17	biimpa 296	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ x = D ) -> y = C )
27	26	exp42 371	. . . . . . . . . . . 12 |- ( ph -> ( x e. A -> ( y e. B -> ( x = D -> y = C ) ) ) )
28	27	com23 78	. . . . . . . . . . 11 |- ( ph -> ( y e. B -> ( x e. A -> ( x = D -> y = C ) ) ) )
29	28	com34 83	. . . . . . . . . 10 |- ( ph -> ( y e. B -> ( x = D -> ( x e. A -> y = C ) ) ) )
30	29	imp32 257	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A -> y = C ) )
31	25, 30	jcai 311	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A /\ y = C ) )
32	22, 31	impbida 600	. . . . . . 7 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
33	32	opabbidv 4181	. . . . . 6 |- ( ph -> { <. y , x >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( y e. B /\ x = D ) } )
34		df-mpt 4178	. . . . . . . . 9 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
35	6, 34	eqtrdi 2283	. . . . . . . 8 |- ( ph -> F = { <. x , y >. | ( x e. A /\ y = C ) } )
36	35	cnveqd 4936	. . . . . . 7 |- ( ph -> `' F = `' { <. x , y >. | ( x e. A /\ y = C ) } )
37		cnvopab 5169	. . . . . . 7 |- `' { <. x , y >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( x e. A /\ y = C ) }
38	36, 37	eqtrdi 2283	. . . . . 6 |- ( ph -> `' F = { <. y , x >. | ( x e. A /\ y = C ) } )
39		df-mpt 4178	. . . . . . 7 |- ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) }
40	39	a1i 9	. . . . . 6 |- ( ph -> ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) } )
41	33, 38, 40	3eqtr4d 2277	. . . . 5 |- ( ph -> `' F = ( y e. B |-> D ) )
42	41	fneq1d 5451	. . . 4 |- ( ph -> ( `' F Fn B <-> ( y e. B |-> D ) Fn B ) )
43	13, 42	mpbird 167	. . 3 |- ( ph -> `' F Fn B )
44		dff1o4 5627	. . 3 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
45	8, 43, 44	sylanbrc 417	. 2 |- ( ph -> F : A -1-1-onto-> B )
46	45, 41	jca 306	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   {copab 4175   |-> cmpt 4176   `'ccnv 4753   Fn wfn 5352   -1-1-onto->wf1o 5356

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364

This theorem is referenced by:  ballotfilemsf1o  13201