Theorem unfilem2 3439

Description: Lemma for proving that the union of two finite sets is finite.

Hypotheses

Ref	Expression
unfilem1.1	|- A e. om
unfilem1.2	|- B e. om
unfilem1.3	|- F = {<.x, y>. | (x e. B /\ y = (A +o x))}

Assertion

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

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem unfilem2

Step	Hyp	Ref	Expression
1		oprex 3018	. . . . . . . . 9 |- (A +o x) e. V
2		unfilem1.3	. . . . . . . . 9 |- F = {<.x, y>. | (x e. B /\ y = (A +o x))}
3	1, 2	fnopab2 2747	. . . . . . . 8 |- F Fn B
4		unfilem1.1	. . . . . . . . 9 |- A e. om
5		unfilem1.2	. . . . . . . . 9 |- B e. om
6	4, 5, 2	unfilem1 3438	. . . . . . . 8 |- ran F = ((A +o B) \ A)
7	3, 6	pm3.2i 234	. . . . . . 7 |- (F Fn B /\ ran F = ((A +o B) \ A))
8		df-fo 2436	. . . . . . 7 |- (F:B-onto->((A +o B) \ A) <-> (F Fn B /\ ran F = ((A +o B) \ A)))
9	7, 8	mpbir 165	. . . . . 6 |- F:B-onto->((A +o B) \ A)
10		fof 2788	. . . . . 6 |- (F:B-onto->((A +o B) \ A) -> F:B-->((A +o B) \ A))
11	9, 10	ax-mp 6	. . . . 5 |- F:B-->((A +o B) \ A)
12		opreq2 3007	. . . . . . . . . 10 |- (x = z -> (A +o x) = (A +o z))
13		oprex 3018	. . . . . . . . . 10 |- (A +o z) e. V
14	12, 2, 13	fvopab4 2871	. . . . . . . . 9 |- (z e. B -> (F` z) = (A +o z))
15		opreq2 3007	. . . . . . . . . 10 |- (x = w -> (A +o x) = (A +o w))
16		oprex 3018	. . . . . . . . . 10 |- (A +o w) e. V
17	15, 2, 16	fvopab4 2871	. . . . . . . . 9 |- (w e. B -> (F` w) = (A +o w))
18	14, 17	cleqan12d 1116	. . . . . . . 8 |- ((z e. B /\ w e. B) -> ((F` z) = (F` w) <-> (A +o z) = (A +o w)))
19		nnacan 3184	. . . . . . . . . 10 |- ((A e. om /\ z e. om /\ w e. om) -> ((A +o z) = (A +o w) <-> z = w))
20	4, 19	mp3an1 639	. . . . . . . . 9 |- ((z e. om /\ w e. om) -> ((A +o z) = (A +o w) <-> z = w))
21		elnn 2383	. . . . . . . . . 10 |- ((z e. B /\ B e. om) -> z e. om)
22	5, 21	mpan2 519	. . . . . . . . 9 |- (z e. B -> z e. om)
23		elnn 2383	. . . . . . . . . 10 |- ((w e. B /\ B e. om) -> w e. om)
24	5, 23	mpan2 519	. . . . . . . . 9 |- (w e. B -> w e. om)
25	20, 22, 24	syl2an 349	. . . . . . . 8 |- ((z e. B /\ w e. B) -> ((A +o z) = (A +o w) <-> z = w))
26	18, 25	bitrd 406	. . . . . . 7 |- ((z e. B /\ w e. B) -> ((F` z) = (F` w) <-> z = w))
27	26	biimpd 135	. . . . . 6 |- ((z e. B /\ w e. B) -> ((F` z) = (F` w) -> z = w))
28	27	rgen2 1248	. . . . 5 |- A.z e. B A.w e. B ((F` z) = (F` w) -> z = w)
29	11, 28	pm3.2i 234	. . . 4 |- (F:B-->((A +o B) \ A) /\ A.z e. B A.w e. B ((F` z) = (F` w) -> z = w))
30		f1fv 2916	. . . 4 |- (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)))
31	29, 30	mpbir 165	. . 3 |- F:B-1-1->((A +o B) \ A)
32	31, 9	pm3.2i 234	. 2 |- (F:B-1-1->((A +o B) \ A) /\ F:B-onto->((A +o B) \ A))
33		df-f1o 2437	. 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)))
34	32, 33	mpbir 165	1 |- F:B-1-1-onto->((A +o B) \ A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201   \ cdif 1484  {copab 2055  omcom 2372  ran crn 2411   Fn wfn 2417  -->wf 2418  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421  ` cfv 2422  (class class class)co 3001   +o coa 3101

This theorem is referenced by:  unfilem3 3440

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

metamath.org