Theorem f1dmex 6273

Description: If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)

Assertion

Ref	Expression
f1dmex	|- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V )

Proof of Theorem f1dmex

Step	Hyp	Ref	Expression
1		f1rn 5540	. . . . 5 |- ( F : A -1-1-> B -> ran F C_ B )
2		ssexg 4226	. . . . 5 |- ( ( ran F C_ B /\ B e. C ) -> ran F e. _V )
3	1, 2	sylan 283	. . . 4 |- ( ( F : A -1-1-> B /\ B e. C ) -> ran F e. _V )
4	3	ex 115	. . 3 |- ( F : A -1-1-> B -> ( B e. C -> ran F e. _V ) )
5		f1cnv 5604	. . . . 5 |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
6		f1ofo 5587	. . . . 5 |- ( `' F : ran F -1-1-onto-> A -> `' F : ran F -onto-> A )
7	5, 6	syl 14	. . . 4 |- ( F : A -1-1-> B -> `' F : ran F -onto-> A )
8		focdmex 6272	. . . 4 |- ( ran F e. _V -> ( `' F : ran F -onto-> A -> A e. _V ) )
9	7, 8	syl5com 29	. . 3 |- ( F : A -1-1-> B -> ( ran F e. _V -> A e. _V ) )
10	4, 9	syld 45	. 2 |- ( F : A -1-1-> B -> ( B e. C -> A e. _V ) )
11	10	imp 124	1 |- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   _Vcvv 2800   C_ wss 3198   `'ccnv 4722   ran crn 4724   -1-1->wf1 5321   -onto->wfo 5322   -1-1-onto->wf1o 5323

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-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

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-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332

This theorem is referenced by:  f1domg  6926