Theorem f1dmex 6311

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 5576	. . . . 5 |- ( F : A -1-1-> B -> ran F C_ B )
2		ssexg 4251	. . . . 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 5640	. . . . 5 |- ( F : A -1-1-> B -> `' F : ran F -1-1-onto-> A )
6		f1ofo 5623	. . . . 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 6310	. . . 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 2205   _Vcvv 2815   C_ wss 3213   `'ccnv 4750   ran crn 4752   -1-1->wf1 5351   -onto->wfo 5352   -1-1-onto->wf1o 5353

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362

This theorem is referenced by:  f1domg  6999