Theorem inresflem 7025

Description: Lemma for inlresf1 7026 and inrresf1 7027. (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 5431	. . 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 5433	. . . 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 2519	. . . 4 |- A. x e. A ( F ` x ) e. B
8		fnfvrnss 5645	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
9	5, 7, 8	mp2an 423	. . 3 |- ran F C_ B
10		df-f 5192	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
11	5, 9, 10	mpbir2an 932	. 2 |- F : A --> B
12		f1ff1 5401	. 2 |- ( ( F : A -1-1-> ( { X } X. A ) /\ F : A --> B ) -> F : A -1-1-> B )
13	3, 11, 12	mp2an 423	1 |- F : A -1-1-> B

Syntax hints:   -> wi 4   e. wcel 2136   A.wral 2444   C_ wss 3116   {csn 3576   X. cxp 4602   ran crn 4605   Fn wfn 5183   -->wf 5184   -1-1->wf1 5185   -1-1-onto->wf1o 5187   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-f1o 5195  df-fv 5196

This theorem is referenced by:  inlresf1  7026  inrresf1  7027