Theorem dff14a 6527

Description: A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)

Assertion

Ref	Expression
dff14a	|- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) ) )

Distinct variable groups:   x, y, A   x, F, y

Allowed substitution hints:   B( x, y)

Proof of Theorem dff14a

Step	Hyp	Ref	Expression
1		dff13 6512	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
2		con34b 306	. . . . 5 |- ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( -. x = y -> -. ( F ` x ) = ( F ` y ) ) )
3		df-ne 2795	. . . . . . 7 |- ( x =/= y <-> -. x = y )
4	3	bicomi 214	. . . . . 6 |- ( -. x = y <-> x =/= y )
5		df-ne 2795	. . . . . . 7 |- ( ( F ` x ) =/= ( F ` y ) <-> -. ( F ` x ) = ( F ` y ) )
6	5	bicomi 214	. . . . . 6 |- ( -. ( F ` x ) = ( F ` y ) <-> ( F ` x ) =/= ( F ` y ) )
7	4, 6	imbi12i 340	. . . . 5 |- ( ( -. x = y -> -. ( F ` x ) = ( F ` y ) ) <-> ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) )
8	2, 7	bitri 264	. . . 4 |- ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) )
9	8	2ralbii 2981	. . 3 |- ( A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) <-> A. x e. A A. y e. A ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) )
10	9	anbi2i 730	. 2 |- ( ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) <-> ( F : A --> B /\ A. x e. A A. y e. A ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) ) )
11	1, 10	bitri 264	1 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( x =/= y -> ( F ` x ) =/= ( F ` y ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   =/= wne 2794   A.wral 2912   -->wf 5884   -1-1->wf1 5885   ` cfv 5888

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896

This theorem is referenced by:  dff14b  6528  pthdlem1  26662  nnfoctbdjlem  40672