Theorem hmeof1o2 15031

Description: A homeomorphism is a 1-1-onto mapping. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
hmeof1o2	|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F : X -1-1-onto-> Y )

Proof of Theorem hmeof1o2

Step	Hyp	Ref	Expression
1		hmeocn 15028	. . . 4 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2		cnf2 14928	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Cn K ) ) -> F : X --> Y )
3	1, 2	syl3an3 1308	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F : X --> Y )
4	3	ffnd 5483	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F Fn X )
5		hmeocnvcn 15029	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
6		cnf2 14928	. . . . 5 |- ( ( K e. (TopOn ` Y ) /\ J e. (TopOn ` X ) /\ `' F e. ( K Cn J ) ) -> `' F : Y --> X )
7	6	3com12 1233	. . . 4 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ `' F e. ( K Cn J ) ) -> `' F : Y --> X )
8	5, 7	syl3an3 1308	. . 3 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> `' F : Y --> X )
9	8	ffnd 5483	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> `' F Fn Y )
10		dff1o4 5591	. 2 |- ( F : X -1-1-onto-> Y <-> ( F Fn X /\ `' F Fn Y ) )
11	4, 9, 10	sylanbrc 417	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( J Homeo K ) ) -> F : X -1-1-onto-> Y )

Syntax hints:   -> wi 4   /\ w3a 1004   e. wcel 2202   `'ccnv 4724   Fn wfn 5321   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326  (class class class)co 6017  TopOnctopon 14733   Cn ccn 14908   Homeochmeo 15023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cn 14911  df-hmeo 15024

This theorem is referenced by:  hmeof1o  15032