Theorem inresflem 7250

Description: Lemma for inlresf1 7251 and inrresf1 7252. (Contributed by BJ, 4-Jul-2022.)

Hypotheses

Ref	Expression
inresflem.1	|- F : A -1-1-onto-> ( { X } X. A )
inresflem.2	|- ( x e. A -> ( F ` x ) e. B )

Assertion

Ref	Expression
inresflem	|- F : A -1-1-> B

Distinct variable groups:   x, A   x, B   x, F

Allowed substitution hint:   X( x)

Proof of Theorem inresflem

Step	Hyp	Ref	Expression
1		inresflem.1	. . 3 |- F : A -1-1-onto-> ( { X } X. A )
2		f1of1 5579	. . 3 |- ( F : A -1-1-onto-> ( { X } X. A ) -> F : A -1-1-> ( { X } X. A ) )
3	1, 2	ax-mp 5	. 2 |- F : A -1-1-> ( { X } X. A )
4		f1ofn 5581	. . . 4 |- ( F : A -1-1-onto-> ( { X } X. A ) -> F Fn A )
5	1, 4	ax-mp 5	. . 3 |- F Fn A
6		inresflem.2	. . . . 5 |- ( x e. A -> ( F ` x ) e. B )
7	6	rgen 2583	. . . 4 |- A. x e. A ( F ` x ) e. B
8		fnfvrnss 5803	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
9	5, 7, 8	mp2an 426	. . 3 |- ran F C_ B
10		df-f 5328	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
11	5, 9, 10	mpbir2an 948	. 2 |- F : A --> B
12		f1ff1 5547	. 2 |- ( ( F : A -1-1-> ( { X } X. A ) /\ F : A --> B ) -> F : A -1-1-> B )
13	3, 11, 12	mp2an 426	1 |- F : A -1-1-> B

Syntax hints:   -> wi 4   e. wcel 2200   A.wral 2508   C_ wss 3198   {csn 3667   X. cxp 4721   ran crn 4724   Fn wfn 5319   -->wf 5320   -1-1->wf1 5321   -1-1-onto->wf1o 5323   ` cfv 5324

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 4205  ax-pow 4262  ax-pr 4297

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-v 2802  df-sbc 3030  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-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-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-f1o 5331  df-fv 5332

This theorem is referenced by:  inlresf1  7251  inrresf1  7252