Theorem oacomf1olem 7644

Description: Lemma for oacomf1o 7645. (Contributed by Mario Carneiro, 30-May-2015.)

Hypothesis

Ref	Expression
oacomf1olem.1	|- F = ( x e. A |-> ( B +o x ) )

Assertion

Ref	Expression
oacomf1olem	|- ( ( A e. On /\ B e. On ) -> ( F : A -1-1-onto-> ran F /\ ( ran F i^i B ) = (/) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   F( x)

Proof of Theorem oacomf1olem

Step	Hyp	Ref	Expression
1		oaf1o 7643	. . . . . . 7 |- ( B e. On -> ( x e. On |-> ( B +o x ) ) : On -1-1-onto-> ( On \ B ) )
2	1	adantl 482	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( x e. On |-> ( B +o x ) ) : On -1-1-onto-> ( On \ B ) )
3		f1of1 6136	. . . . . 6 |- ( ( x e. On |-> ( B +o x ) ) : On -1-1-onto-> ( On \ B ) -> ( x e. On |-> ( B +o x ) ) : On -1-1-> ( On \ B ) )
4	2, 3	syl 17	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( x e. On |-> ( B +o x ) ) : On -1-1-> ( On \ B ) )
5		onss 6990	. . . . . 6 |- ( A e. On -> A C_ On )
6	5	adantr 481	. . . . 5 |- ( ( A e. On /\ B e. On ) -> A C_ On )
7		f1ssres 6108	. . . . 5 |- ( ( ( x e. On |-> ( B +o x ) ) : On -1-1-> ( On \ B ) /\ A C_ On ) -> ( ( x e. On |-> ( B +o x ) ) |` A ) : A -1-1-> ( On \ B ) )
8	4, 6, 7	syl2anc 693	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( ( x e. On |-> ( B +o x ) ) |` A ) : A -1-1-> ( On \ B ) )
9	6	resmptd 5452	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( ( x e. On |-> ( B +o x ) ) |` A ) = ( x e. A |-> ( B +o x ) ) )
10		oacomf1olem.1	. . . . . 6 |- F = ( x e. A |-> ( B +o x ) )
11	9, 10	syl6eqr 2674	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( ( x e. On |-> ( B +o x ) ) |` A ) = F )
12		f1eq1 6096	. . . . 5 |- ( ( ( x e. On |-> ( B +o x ) ) |` A ) = F -> ( ( ( x e. On |-> ( B +o x ) ) |` A ) : A -1-1-> ( On \ B ) <-> F : A -1-1-> ( On \ B ) ) )
13	11, 12	syl 17	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( ( ( x e. On |-> ( B +o x ) ) |` A ) : A -1-1-> ( On \ B ) <-> F : A -1-1-> ( On \ B ) ) )
14	8, 13	mpbid 222	. . 3 |- ( ( A e. On /\ B e. On ) -> F : A -1-1-> ( On \ B ) )
15		f1f1orn 6148	. . 3 |- ( F : A -1-1-> ( On \ B ) -> F : A -1-1-onto-> ran F )
16	14, 15	syl 17	. 2 |- ( ( A e. On /\ B e. On ) -> F : A -1-1-onto-> ran F )
17		f1f 6101	. . . 4 |- ( F : A -1-1-> ( On \ B ) -> F : A --> ( On \ B ) )
18		frn 6053	. . . 4 |- ( F : A --> ( On \ B ) -> ran F C_ ( On \ B ) )
19	14, 17, 18	3syl 18	. . 3 |- ( ( A e. On /\ B e. On ) -> ran F C_ ( On \ B ) )
20	19	difss2d 3740	. . . 4 |- ( ( A e. On /\ B e. On ) -> ran F C_ On )
21		reldisj 4020	. . . 4 |- ( ran F C_ On -> ( ( ran F i^i B ) = (/) <-> ran F C_ ( On \ B ) ) )
22	20, 21	syl 17	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( ran F i^i B ) = (/) <-> ran F C_ ( On \ B ) ) )
23	19, 22	mpbird 247	. 2 |- ( ( A e. On /\ B e. On ) -> ( ran F i^i B ) = (/) )
24	16, 23	jca 554	1 |- ( ( A e. On /\ B e. On ) -> ( F : A -1-1-onto-> ran F /\ ( ran F i^i B ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   ran crn 5115   |` cres 5116   Oncon0 5723   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887  (class class class)co 6650   +o coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  oacomf1o  7645  onacda  9019