Theorem unfilem2 8225

Description: Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
unfilem1.1	|- A e. om
unfilem1.2	|- B e. om
unfilem1.3	|- F = ( x e. B |-> ( A +o x ) )

Assertion

Ref	Expression
unfilem2	|- F : B -1-1-onto-> ( ( A +o B ) \ A )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   F( x)

Proof of Theorem unfilem2

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . . . . 6 |- ( A +o x ) e. _V
2		unfilem1.3	. . . . . 6 |- F = ( x e. B |-> ( A +o x ) )
3	1, 2	fnmpti 6022	. . . . 5 |- F Fn B
4		unfilem1.1	. . . . . 6 |- A e. om
5		unfilem1.2	. . . . . 6 |- B e. om
6	4, 5, 2	unfilem1 8224	. . . . 5 |- ran F = ( ( A +o B ) \ A )
7		df-fo 5894	. . . . 5 |- ( F : B -onto-> ( ( A +o B ) \ A ) <-> ( F Fn B /\ ran F = ( ( A +o B ) \ A ) ) )
8	3, 6, 7	mpbir2an 955	. . . 4 |- F : B -onto-> ( ( A +o B ) \ A )
9		fof 6115	. . . 4 |- ( F : B -onto-> ( ( A +o B ) \ A ) -> F : B --> ( ( A +o B ) \ A ) )
10	8, 9	ax-mp 5	. . 3 |- F : B --> ( ( A +o B ) \ A )
11		oveq2 6658	. . . . . . . 8 |- ( x = z -> ( A +o x ) = ( A +o z ) )
12		ovex 6678	. . . . . . . 8 |- ( A +o z ) e. _V
13	11, 2, 12	fvmpt 6282	. . . . . . 7 |- ( z e. B -> ( F ` z ) = ( A +o z ) )
14		oveq2 6658	. . . . . . . 8 |- ( x = w -> ( A +o x ) = ( A +o w ) )
15		ovex 6678	. . . . . . . 8 |- ( A +o w ) e. _V
16	14, 2, 15	fvmpt 6282	. . . . . . 7 |- ( w e. B -> ( F ` w ) = ( A +o w ) )
17	13, 16	eqeqan12d 2638	. . . . . 6 |- ( ( z e. B /\ w e. B ) -> ( ( F ` z ) = ( F ` w ) <-> ( A +o z ) = ( A +o w ) ) )
18		elnn 7075	. . . . . . . 8 |- ( ( z e. B /\ B e. om ) -> z e. om )
19	5, 18	mpan2 707	. . . . . . 7 |- ( z e. B -> z e. om )
20		elnn 7075	. . . . . . . 8 |- ( ( w e. B /\ B e. om ) -> w e. om )
21	5, 20	mpan2 707	. . . . . . 7 |- ( w e. B -> w e. om )
22		nnacan 7708	. . . . . . . 8 |- ( ( A e. om /\ z e. om /\ w e. om ) -> ( ( A +o z ) = ( A +o w ) <-> z = w ) )
23	4, 22	mp3an1 1411	. . . . . . 7 |- ( ( z e. om /\ w e. om ) -> ( ( A +o z ) = ( A +o w ) <-> z = w ) )
24	19, 21, 23	syl2an 494	. . . . . 6 |- ( ( z e. B /\ w e. B ) -> ( ( A +o z ) = ( A +o w ) <-> z = w ) )
25	17, 24	bitrd 268	. . . . 5 |- ( ( z e. B /\ w e. B ) -> ( ( F ` z ) = ( F ` w ) <-> z = w ) )
26	25	biimpd 219	. . . 4 |- ( ( z e. B /\ w e. B ) -> ( ( F ` z ) = ( F ` w ) -> z = w ) )
27	26	rgen2a 2977	. . 3 |- A. z e. B A. w e. B ( ( F ` z ) = ( F ` w ) -> z = w )
28		dff13 6512	. . 3 |- ( F : B -1-1-> ( ( A +o B ) \ A ) <-> ( F : B --> ( ( A +o B ) \ A ) /\ A. z e. B A. w e. B ( ( F ` z ) = ( F ` w ) -> z = w ) ) )
29	10, 27, 28	mpbir2an 955	. 2 |- F : B -1-1-> ( ( A +o B ) \ A )
30		df-f1o 5895	. 2 |- ( F : B -1-1-onto-> ( ( A +o B ) \ A ) <-> ( F : B -1-1-> ( ( A +o B ) \ A ) /\ F : B -onto-> ( ( A +o B ) \ A ) ) )
31	29, 8, 30	mpbir2an 955	1 |- F : B -1-1-onto-> ( ( A +o B ) \ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   omcom 7065   +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-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-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:  unfilem3  8226