Theorem inresflem 7188

Description: Lemma for inlresf1 7189 and inrresf1 7190. (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 5543	. . 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 5545	. . . 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 2561	. . . 4 |- A. x e. A ( F ` x ) e. B
8		fnfvrnss 5763	. . . 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 5294	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
11	5, 9, 10	mpbir2an 945	. 2 |- F : A --> B
12		f1ff1 5511	. 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 2178   A.wral 2486   C_ wss 3174   {csn 3643   X. cxp 4691   ran crn 4694   Fn wfn 5285   -->wf 5286   -1-1->wf1 5287   -1-1-onto->wf1o 5289   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-f1o 5297  df-fv 5298

This theorem is referenced by:  inlresf1  7189  inrresf1  7190