Theorem f1ofveu 6645

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 6149	. . . 4 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
2		f1of 6137	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
3	1, 2	syl 17	. . 3 |- ( F : A -1-1-onto-> B -> `' F : B --> A )
4		feu 6080	. . 3 |- ( ( `' F : B --> A /\ C e. B ) -> E! x e. A <. C , x >. e. `' F )
5	3, 4	sylan 488	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A <. C , x >. e. `' F )
6		f1ocnvfvb 6535	. . . . . 6 |- ( ( F : A -1-1-onto-> B /\ x e. A /\ C e. B ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
7	6	3com23 1271	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> ( `' F ` C ) = x ) )
8		dff1o4 6145	. . . . . . 7 |- ( F : A -1-1-onto-> B <-> ( F Fn A /\ `' F Fn B ) )
9	8	simprbi 480	. . . . . 6 |- ( F : A -1-1-onto-> B -> `' F Fn B )
10		fnopfvb 6237	. . . . . . 7 |- ( ( `' F Fn B /\ C e. B ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
11	10	3adant3 1081	. . . . . 6 |- ( ( `' F Fn B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
12	9, 11	syl3an1 1359	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( `' F ` C ) = x <-> <. C , x >. e. `' F ) )
13	7, 12	bitrd 268	. . . 4 |- ( ( F : A -1-1-onto-> B /\ C e. B /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
14	13	3expa 1265	. . 3 |- ( ( ( F : A -1-1-onto-> B /\ C e. B ) /\ x e. A ) -> ( ( F ` x ) = C <-> <. C , x >. e. `' F ) )
15	14	reubidva 3125	. 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 247	1 |- ( ( F : A -1-1-onto-> B /\ C e. B ) -> E! x e. A ( F ` x ) = C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E!wreu 2914   <.cop 4183   `'ccnv 5113   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` 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-8 1992  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-pow 4843  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-reu 2919  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  1arith2  15632  disjrdx  29404