Theorem f1ofveu 6066

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 5650	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1of 5637	. . . 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 5572	. . 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 5979	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
7	6	3com23 1240	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
8		dff1o4 5645	. . . . . . 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 5739	. . . . . . 7 |- ( ( `' F Fn B /\ C e. B ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
11	10	3adant3 1048	. . . . . 6 |- ( ( `' F Fn B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
12	9, 11	syl3an1 1311	. . . . 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 1234	. . 3 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
15	14	reubidva 2736	. 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 1009   = wceq 1402   e. wcel 2209   E!wreu 2530   <.cop 3711   `'ccnv 4771   Fn wfn 5370   -->wf 5371   -1-1-onto->wf1o 5374   ` cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383

This theorem is referenced by:  1arith2  13128  uspgredgiedg  16336