Theorem f1od 6052

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 6051	. 2 |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
6	5	simpld 111	1 |- ( ph -> F : A -1-1-onto-> B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   |-> cmpt 4050   `'ccnv 4610   -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:  cnvf1o  6204  ixpsnf1o  6714  en2d  6746  mptfzshft  11405  fsumrev  11406  fprodrev  11582