Theorem f1ocnvd 6051

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 2543	. . . 4 |- ( ph -> A. x e. A C e. W )
3		f1od.1	. . . . 5 |- F = ( x e. A |-> C )
4	3	fnmpt 5324	. . . 4 |- ( A. x e. A C e. W -> F Fn A )
5	2, 4	syl 14	. . 3 |- ( ph -> F Fn A )
6		f1od.3	. . . . . 6 |- ( ( ph /\ y e. B ) -> D e. X )
7	6	ralrimiva 2543	. . . . 5 |- ( ph -> A. y e. B D e. X )
8		eqid 2170	. . . . . 6 |- ( y e. B |-> D ) = ( y e. B |-> D )
9	8	fnmpt 5324	. . . . 5 |- ( A. y e. B D e. X -> ( y e. B |-> D ) Fn B )
10	7, 9	syl 14	. . . 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 4055	. . . . . 6 |- ( ph -> { <. y , x >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( y e. B /\ x = D ) } )
13		df-mpt 4052	. . . . . . . . 9 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
14	3, 13	eqtri 2191	. . . . . . . 8 |- F = { <. x , y >. | ( x e. A /\ y = C ) }
15	14	cnveqi 4786	. . . . . . 7 |- `' F = `' { <. x , y >. | ( x e. A /\ y = C ) }
16		cnvopab 5012	. . . . . . 7 |- `' { <. x , y >. | ( x e. A /\ y = C ) } = { <. y , x >. | ( x e. A /\ y = C ) }
17	15, 16	eqtri 2191	. . . . . 6 |- `' F = { <. y , x >. | ( x e. A /\ y = C ) }
18		df-mpt 4052	. . . . . 6 |- ( y e. B |-> D ) = { <. y , x >. | ( y e. B /\ x = D ) }
19	12, 17, 18	3eqtr4g 2228	. . . . 5 |- ( ph -> `' F = ( y e. B |-> D ) )
20	19	fneq1d 5288	. . . 4 |- ( ph -> ( `' F Fn B <-> ( y e. B |-> D ) Fn B ) )
21	10, 20	mpbird 166	. . 3 |- ( ph -> `' F Fn B )
22		dff1o4 5450	. . 3 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
23	5, 21, 22	sylanbrc 415	. 2 |- ( ph -> F : A -1-1-onto-> B )
24	23, 19	jca 304	1 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   {copab 4049   |-> cmpt 4050   `'ccnv 4610   Fn wfn 5193   -1-1-onto->wf1o 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1od  6052  f1ocnv2d  6053