Theorem f1ofveu 5755

Description: There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.)

Assertion

Ref	Expression
f1ofveu	|- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A ( F ` x ) = C )

Distinct variable groups:   x, A   x, B   x, C   x, F

Proof of Theorem f1ofveu

Step	Hyp	Ref	Expression
1		f1ocnv 5373	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1of 5360	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
3	1, 2	syl 14	. . 3 |- ( F : A -1-1-onto-> B -> `' F : B --> A )
4		feu 5300	. . 3 |- ( ( `' F : B --> A /\ C e. B ) -> E! x e. A <. C , x >. e. `' F )
5	3, 4	sylan 281	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A <. C , x >. e. `' F )
6		f1ocnvfvb 5674	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
7	6	3com23 1187	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
8		dff1o4 5368	. . . . . . 7 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
9	8	simprbi 273	. . . . . 6 |- ( F : A -1-1-onto-> B -> `' F Fn B )
10		fnopfvb 5456	. . . . . . 7 |- ( ( `' F Fn B /\ C e. B ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
11	10	3adant3 1001	. . . . . 6 |- ( ( `' F Fn B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
12	9, 11	syl3an1 1249	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
13	7, 12	bitrd 187	. . . 4 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
14	13	3expa 1181	. . 3 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
15	14	reubidva 2611	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> ( E! x e. A ( F ` x ) = C <-> E! x e. A <. C , x >. e. `' F ) )
16	5, 15	mpbird 166	1 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A ( F ` x ) = C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E!wreu 2416   <.cop 3525   `'ccnv 4533   Fn wfn 5113   -->wf 5114   -1-1-onto->wf1o 5117   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126

This theorem is referenced by: (None)