Theorem hmeof1o 20434

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 20430	. . 3 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2		cntop1 19911	. . . . 5 |- ( F e. ( J Cn K ) -> J e. Top )
3		hmeof1o.1	. . . . . 6 |- X = U. J
4	3	toptopon 19604	. . . . 5 |- ( J e. Top <-> J e. (TopOn ` X ) )
5	2, 4	sylib 196	. . . 4 |- ( F e. ( J Cn K ) -> J e. (TopOn ` X ) )
6		cntop2 19912	. . . . 5 |- ( F e. ( J Cn K ) -> K e. Top )
7		hmeof1o.2	. . . . . 6 |- Y = U. K
8	7	toptopon 19604	. . . . 5 |- ( K e. Top <-> K e. (TopOn ` Y ) )
9	6, 8	sylib 196	. . . 4 |- ( F e. ( J Cn K ) -> K e. (TopOn ` Y ) )
10	5, 9	jca 530	. . 3 |- ( F e. ( J Cn K ) -> ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) )
11	1, 10	syl 16	. 2 |- ( F e. ( J Homeo K ) -> ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) ) )
12		hmeof1o2 20433	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F : X -1-1-onto-> Y )
13	12	3expia 1196	. 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 367   = wceq 1398   e. wcel 1823   U.cuni 4235   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   Topctop 19564  TopOnctopon 19565   Cn ccn 19895   Homeochmeo 20423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19569  df-topon 19572  df-cn 19898  df-hmeo 20425

This theorem is referenced by:  hmeoopn  20436  hmeocld  20437  hmeontr  20439  hmeoimaf1o  20440  hmeoqtop  20445  haushmphlem  20457  cmphmph  20458  conhmph  20459  reghmph  20463  nrmhmph  20464  hmphdis  20466  hmphen2  20469  cmphaushmeo  20470  txhmeo  20473  tpr2rico  28132  mndpluscn  28146