Theorem f1o3d 29431

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 2966	. . . . 5 |- ( ph -> A. x e. A C e. B )
3		eqid 2622	. . . . . 6 |- ( x e. A |-> C ) = ( x e. A |-> C )
4	3	fnmpt 6020	. . . . 5 |- ( A. x e. A C e. B -> ( x e. A |-> C ) Fn A )
5	2, 4	syl 17	. . . 4 |- ( ph -> ( x e. A |-> C ) Fn A )
6		f1o3d.1	. . . . 5 |- ( ph -> F = ( x e. A |-> C ) )
7	6	fneq1d 5981	. . . 4 |- ( ph -> ( F Fn A <-> ( x e. A |-> C ) Fn A ) )
8	5, 7	mpbird 247	. . 3 |- ( ph -> F Fn A )
9		f1o3d.3	. . . . . 6 |- ( ( ph /\ y e. B ) -> D e. A )
10	9	ralrimiva 2966	. . . . 5 |- ( ph -> A. y e. B D e. A )
11		eqid 2622	. . . . . 6 |- ( y e. B |-> D ) = ( y e. B |-> D )
12	11	fnmpt 6020	. . . . 5 |- ( A. y e. B D e. A -> ( y e. B |-> D ) Fn B )
13	10, 12	syl 17	. . . 4 |- ( ph -> ( y e. B |-> D ) Fn B )
14		eleq1a 2696	. . . . . . . . . . 11 |- ( C e. B -> ( y = C -> y e. B ) )
15	1, 14	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( y = C -> y e. B ) )
16	15	impr 649	. . . . . . . . 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 502	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ y = C ) -> x = D )
19	18	exp42 639	. . . . . . . . . . 11 |- ( ph -> ( x e. A -> ( y e. B -> ( y = C -> x = D ) ) ) )
20	19	com34 91	. . . . . . . . . 10 |- ( ph -> ( x e. A -> ( y = C -> ( y e. B -> x = D ) ) ) )
21	20	imp32 449	. . . . . . . . 9 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B -> x = D ) )
22	16, 21	jcai 559	. . . . . . . 8 |- ( ( ph /\ ( x e. A /\ y = C ) ) -> ( y e. B /\ x = D ) )
23		eleq1a 2696	. . . . . . . . . . 11 |- ( D e. A -> ( x = D -> x e. A ) )
24	9, 23	syl 17	. . . . . . . . . 10 |- ( ( ph /\ y e. B ) -> ( x = D -> x e. A ) )
25	24	impr 649	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> x e. A )
26	17	biimpa 501	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x e. A /\ y e. B ) ) /\ x = D ) -> y = C )
27	26	exp42 639	. . . . . . . . . . . 12 |- ( ph -> ( x e. A -> ( y e. B -> ( x = D -> y = C ) ) ) )
28	27	com23 86	. . . . . . . . . . 11 |- ( ph -> ( y e. B -> ( x e. A -> ( x = D -> y = C ) ) ) )
29	28	com34 91	. . . . . . . . . 10 |- ( ph -> ( y e. B -> ( x = D -> ( x e. A -> y = C ) ) ) )
30	29	imp32 449	. . . . . . . . 9 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A -> y = C ) )
31	25, 30	jcai 559	. . . . . . . 8 |- ( ( ph /\ ( y e. B /\ x = D ) ) -> ( x e. A /\ y = C ) )
32	22, 31	impbida 877	. . . . . . 7 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
33	32	opabbidv 4716	. . . . . 6 |- ( ph -> { <. y , x >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( y e. B /\ x = D ) } )
34		df-mpt 4730	. . . . . . . . 9 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
35	6, 34	syl6eq 2672	. . . . . . . 8 |- ( ph -> F = { <. x , y >. | ( x e. A /\ y = C ) } )
36	35	cnveqd 5298	. . . . . . 7 |- ( ph -> `' F = `' { <. x , y >. | ( x e. A /\ y = C ) } )
37		cnvopab 5533	. . . . . . 7 |- `' { <. x , y >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( x e. A /\ y = C ) }
38	36, 37	syl6eq 2672	. . . . . 6 |- ( ph -> `' F = { <. y , x >. | ( x e. A /\ y = C ) } )
39		df-mpt 4730	. . . . . . 7 |- ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) }
40	39	a1i 11	. . . . . 6 |- ( ph -> ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) } )
41	33, 38, 40	3eqtr4d 2666	. . . . 5 |- ( ph -> `' F = ( y e. B |-> D ) )
42	41	fneq1d 5981	. . . 4 |- ( ph -> ( `' F Fn B <-> ( y e. B |-> D ) Fn B ) )
43	13, 42	mpbird 247	. . 3 |- ( ph -> `' F Fn B )
44		dff1o4 6145	. . 3 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
45	8, 43, 44	sylanbrc 698	. 2 |- ( ph -> F : A -1-1-onto-> B )
46	45, 41	jca 554	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {copab 4712   |-> cmpt 4729   `'ccnv 5113   Fn wfn 5883   -1-1-onto->wf1o 5887

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  fmptco1f1o  29434  ballotlemsf1o  30575