Theorem hmeof1o 20856

Description: A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Hypotheses

Ref	Expression
hmeof1o.1	|- X = U. J
hmeof1o.2	|- Y = U. K

Assertion

Ref	Expression
hmeof1o	|- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> Y )

Proof of Theorem hmeof1o

Step	Hyp	Ref	Expression
1		hmeocn 20852	. . 3 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2		cntop1 20333	. . . . 5 |- ( F e. ( J Cn K ) -> J e. Top )
3		hmeof1o.1	. . . . . 6 |- X = U. J
4	3	toptopon 20025	. . . . 5 |- ( J e. Top <-> J e. (TopOn ` X ) )
5	2, 4	sylib 201	. . . 4 |- ( F e. ( J Cn K ) -> J e. (TopOn ` X ) )
6		cntop2 20334	. . . . 5 |- ( F e. ( J Cn K ) -> K e. Top )
7		hmeof1o.2	. . . . . 6 |- Y = U. K
8	7	toptopon 20025	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` Y ) )
9	6, 8	sylib 201	. . . 4 |- ( F e. ( J Cn K ) -> K e. (TopOn ` Y ) )
10	5, 9	jca 541	. . 3 |- ( F e. ( J Cn K ) -> ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) )
11	1, 10	syl 17	. 2 |- ( F e. ( J Homeo K ) -> ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) )
12		hmeof1o2 20855	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F : X -1-1-onto-> Y )
13	12	3expia 1233	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) -> ( F e. ( J Homeo K ) -> F : X -1-1-onto-> Y ) )
14	11, 13	mpcom 36	1 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> Y )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   U.cuni 4190   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   Homeochmeo 20845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-top 19998  df-topon 20000  df-cn 20320  df-hmeo 20847

This theorem is referenced by:  hmeoopn  20858  hmeocld  20859  hmeontr  20861  hmeoimaf1o  20862  hmeoqtop  20867  haushmphlem  20879  cmphmph  20880  conhmph  20881  reghmph  20885  nrmhmph  20886  hmphdis  20888  hmphen2  20891  cmphaushmeo  20892  txhmeo  20895  tpr2rico  28792  mndpluscn  28806