Theorem f1od 6139

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)

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
f1od	|- ( ph -> F : A -1-1-onto-> B )

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 f1od

Step	Hyp	Ref	Expression
1		f1od.1	. . 3 |- F = ( x e. A |-> C )
2		f1od.2	. . 3 |- ( ( ph /\ x e. A ) -> C e. W )
3		f1od.3	. . 3 |- ( ( ph /\ y e. B ) -> D e. X )
4		f1od.4	. . 3 |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )
5	1, 2, 3, 4	f1ocnvd 6138	. 2 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
6	5	simpld 112	1 |- ( ph -> F : A -1-1-onto-> B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   |-> cmpt 4104   `'ccnv 4672   -1-1-onto->wf1o 5267

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275

This theorem is referenced by:  cnvf1o  6301  ixpsnf1o  6813  en2d  6845  pw2f1odc  6914  seqf1oglem1  10645  mptfzshft  11672  fsumrev  11673  fprodrev  11849