Theorem inresflem 7037

Description: Lemma for inlresf1 7038 and inrresf1 7039. (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 5441	. . 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 5443	. . . 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 2523	. . . 4 |- A. x e. A ( F ` x ) e. B
8		fnfvrnss 5656	. . . 4 |- ( ( F Fn A /\ A. x e. A ( F ` x ) e. B ) -> ran F C_ B )
9	5, 7, 8	mp2an 424	. . 3 |- ran F C_ B
10		df-f 5202	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
11	5, 9, 10	mpbir2an 937	. 2 |- F : A --> B
12		f1ff1 5411	. 2 |- ( ( F : A -1-1-> ( { X } X. A ) /\ F : A --> B ) -> F : A -1-1-> B )
13	3, 11, 12	mp2an 424	1 |- F : A -1-1-> B

Syntax hints:   -> wi 4   e. wcel 2141   A.wral 2448   C_ wss 3121   {csn 3583   X. cxp 4609   ran crn 4612   Fn wfn 5193   -->wf 5194   -1-1->wf1 5195   -1-1-onto->wf1o 5197   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-f1o 5205  df-fv 5206

This theorem is referenced by:  inlresf1  7038  inrresf1  7039