Theorem f1ofveu 5989

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 5585	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1of 5572	. . . 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 5508	. . 3 |- ( ( `' F : B --> A /\ C e. B ) -> E! x e. A <. C , x >. e. `' F )
5	3, 4	sylan 283	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A <. C , x >. e. `' F )
6		f1ocnvfvb 5904	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
7	6	3com23 1233	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
8		dff1o4 5580	. . . . . . 7 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
9	8	simprbi 275	. . . . . 6 |- ( F : A -1-1-onto-> B -> `' F Fn B )
10		fnopfvb 5673	. . . . . . 7 |- ( ( `' F Fn B /\ C e. B ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
11	10	3adant3 1041	. . . . . 6 |- ( ( `' F Fn B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
12	9, 11	syl3an1 1304	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
13	7, 12	bitrd 188	. . . 4 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
14	13	3expa 1227	. . 3 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
15	14	reubidva 2715	. 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 167	1 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A ( F ` x ) = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E!wreu 2510   <.cop 3669   `'ccnv 4718   Fn wfn 5313   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326

This theorem is referenced by:  1arith2  12891  uspgredgiedg  15976