Theorem f1ocnvd 6884

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
f1od.1	|- F = ( x e. A |-> C )
f1od.2	|- ( ( ph /\ x e. A ) -> C e. W )
f1od.3	|- ( ( ph /\ y e. B ) -> D e. X )
f1od.4	|- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )

Assertion

Ref	Expression
f1ocnvd	|- ( 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)   W( x, y)   X( x, y)

Proof of Theorem f1ocnvd

Step	Hyp	Ref	Expression
1		f1od.2	. . . . 5 |- ( ( ph /\ x e. A ) -> C e. W )
2	1	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. A C e. W )
3		f1od.1	. . . . 5 |- F = ( x e. A |-> C )
4	3	fnmpt 6020	. . . 4 |- ( A. x e. A C e. W -> F Fn A )
5	2, 4	syl 17	. . 3 |- ( ph -> F Fn A )
6		f1od.3	. . . . . 6 |- ( ( ph /\ y e. B ) -> D e. X )
7	6	ralrimiva 2966	. . . . 5 |- ( ph -> A. y e. B D e. X )
8		eqid 2622	. . . . . 6 |- ( y e. B |-> D ) = ( y e. B |-> D )
9	8	fnmpt 6020	. . . . 5 |- ( A. y e. B D e. X -> ( y e. B |-> D ) Fn B )
10	7, 9	syl 17	. . . 4 |- ( ph -> ( y e. B |-> D ) Fn B )
11		f1od.4	. . . . . . 7 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
12	11	opabbidv 4716	. . . . . 6 |- ( ph -> { <. y , x >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( y e. B /\ x = D ) } )
13		df-mpt 4730	. . . . . . . . 9 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
14	3, 13	eqtri 2644	. . . . . . . 8 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
15	14	cnveqi 5297	. . . . . . 7 |- `' F = `' { <. x , y >. | ( x e. A /\ y = C ) }
16		cnvopab 5533	. . . . . . 7 |- `' { <. x , y >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( x e. A /\ y = C ) }
17	15, 16	eqtri 2644	. . . . . 6 |- `' F = { <. y , x >. | ( x e. A /\ y = C ) }
18		df-mpt 4730	. . . . . 6 |- ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) }
19	12, 17, 18	3eqtr4g 2681	. . . . 5 |- ( ph -> `' F = ( y e. B |-> D ) )
20	19	fneq1d 5981	. . . 4 |- ( ph -> ( `' F Fn B <-> ( y e. B |-> D ) Fn B ) )
21	10, 20	mpbird 247	. . 3 |- ( ph -> `' F Fn B )
22		dff1o4 6145	. . 3 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
23	5, 21, 22	sylanbrc 698	. 2 |- ( ph -> F : A -1-1-onto-> B )
24	23, 19	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:  f1od  6885  f1ocnv2d  6886  pw2f1ocnv  37604