Theorem enf1f1o 15720

Description: A one-to-one mapping of finite sets with the same cardinality is bijective.

Assertion

Ref	Expression
enf1f1o	|- ((A e. Fin /\ B ~~ A) -> (F:A-1-1->B -> F:A-1-1-onto->B))

Proof of Theorem enf1f1o

Step	Hyp	Ref	Expression
1		dff1o5 4646	. . 3 |- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))
2		simpr 350	. . 3 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> F:A-1-1->B)
3		f1f1orn 4649	. . . . 5 |- (F:A-1-1->B -> F:A-1-1-onto->ran F)
4	3	adantl 424	. . . 4 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> F:A-1-1-onto->ran F)
5		f1oeng 5454	. . . . . . 7 |- ((A e. Fin /\ F:A-1-1-onto->ran F) -> A ~~ ran F)
6	5	ex 402	. . . . . 6 |- (A e. Fin -> (F:A-1-1-onto->ran F -> A ~~ ran F))
7	6	ad2antrr 440	. . . . 5 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> (F:A-1-1-onto->ran F -> A ~~ ran F))
8		ssexg 3457	. . . . . . . . . . . . 13 |- ((ran F C_ B /\ B e. Fin) -> ran F e. _V)
9		ensymg 5470	. . . . . . . . . . . . 13 |- (ran F e. _V -> (B ~~ ran F -> ran F ~~ B))
10	8, 9	syl 12	. . . . . . . . . . . 12 |- ((ran F C_ B /\ B e. Fin) -> (B ~~ ran F -> ran F ~~ B))
11	10	ancoms 484	. . . . . . . . . . 11 |- ((B e. Fin /\ ran F C_ B) -> (B ~~ ran F -> ran F ~~ B))
12		sfseqeq 10169	. . . . . . . . . . . 12 |- ((B e. Fin /\ ran F C_ B /\ ran F ~~ B) -> ran F = B)
13	12	3expia 1069	. . . . . . . . . . 11 |- ((B e. Fin /\ ran F C_ B) -> (ran F ~~ B -> ran F = B))
14	11, 13	syld 30	. . . . . . . . . 10 |- ((B e. Fin /\ ran F C_ B) -> (B ~~ ran F -> ran F = B))
15		entr 5473	. . . . . . . . . 10 |- ((B ~~ A /\ A ~~ ran F) -> B ~~ ran F)
16	14, 15	syl5 20	. . . . . . . . 9 |- ((B e. Fin /\ ran F C_ B) -> ((B ~~ A /\ A ~~ ran F) -> ran F = B))
17	16	expdimp 406	. . . . . . . 8 |- (((B e. Fin /\ ran F C_ B) /\ B ~~ A) -> (A ~~ ran F -> ran F = B))
18	17	an1rs 547	. . . . . . 7 |- (((B e. Fin /\ B ~~ A) /\ ran F C_ B) -> (A ~~ ran F -> ran F = B))
19		f1f 4610	. . . . . . . 8 |- (F:A-1-1->B -> F:A-->B)
20		frn 4569	. . . . . . . 8 |- (F:A-->B -> ran F C_ B)
21	19, 20	syl 12	. . . . . . 7 |- (F:A-1-1->B -> ran F C_ B)
22	18, 21	sylan2 500	. . . . . 6 |- (((B e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> (A ~~ ran F -> ran F = B))
23		enfi 5627	. . . . . . . . . . 11 |- ((A e. Fin /\ B ~~ A) -> (B e. Fin <-> A e. Fin))
24	23	exbiri 421	. . . . . . . . . 10 |- (A e. Fin -> (B ~~ A -> (A e. Fin -> B e. Fin)))
25	24	com23 36	. . . . . . . . 9 |- (A e. Fin -> (A e. Fin -> (B ~~ A -> B e. Fin)))
26	25	imp 377	. . . . . . . 8 |- ((A e. Fin /\ A e. Fin) -> (B ~~ A -> B e. Fin))
27	26	anidms 480	. . . . . . 7 |- (A e. Fin -> (B ~~ A -> B e. Fin))
28	27	impac 423	. . . . . 6 |- ((A e. Fin /\ B ~~ A) -> (B e. Fin /\ B ~~ A))
29	22, 28	sylan 497	. . . . 5 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> (A ~~ ran F -> ran F = B))
30	7, 29	syld 30	. . . 4 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> (F:A-1-1-onto->ran F -> ran F = B))
31	4, 30	mpd 29	. . 3 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> ran F = B)
32	1, 2, 31	sylanbrc 527	. 2 |- (((A e. Fin /\ B ~~ A) /\ F:A-1-1->B) -> F:A-1-1-onto->B)
33	32	ex 402	1 |- ((A e. Fin /\ B ~~ A) -> (F:A-1-1->B -> F:A-1-1-onto->B))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  ran crn 3987  -->wf 3994  -1-1->wf1 3995  -1-1-onto->wf1o 3997   ~~ cen 5423  Fincfn 5426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430

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