Theorem sshauslem 21176

Description: Lemma for sshaus 21179 and similar theorems. If the topological property A is preserved under injective preimages, then a topology finer than one with property A also has property A. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
t1sep.1	|- X = U. J
sshauslem.2	|- ( J e. A -> J e. Top )
sshauslem.3	|- ( ( J e. A /\ ( _I |` X ) : X -1-1-> X /\ ( _I |` X ) e. ( K Cn J ) ) -> K e. A )

Assertion

Ref	Expression
sshauslem	|- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> K e. A )

Proof of Theorem sshauslem

Step	Hyp	Ref	Expression
1		simp1 1061	. 2 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> J e. A )
2		f1oi 6174	. . 3 |- ( _I |` X ) : X -1-1-onto-> X
3		f1of1 6136	. . 3 |- ( ( _I |` X ) : X -1-1-onto-> X -> ( _I |` X ) : X -1-1-> X )
4	2, 3	mp1i 13	. 2 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( _I |` X ) : X -1-1-> X )
5		simp3 1063	. . 3 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> J C_ K )
6		simp2 1062	. . . 4 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> K e. (TopOn ` X ) )
7		sshauslem.2	. . . . . 6 |- ( J e. A -> J e. Top )
8	7	3ad2ant1 1082	. . . . 5 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> J e. Top )
9		t1sep.1	. . . . . 6 |- X = U. J
10	9	toptopon 20722	. . . . 5 |- ( J e. Top <-> J e. (TopOn ` X ) )
11	8, 10	sylib 208	. . . 4 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> J e. (TopOn ` X ) )
12		ssidcn 21059	. . . 4 |- ( ( K e. (TopOn ` X ) /\ J e. (TopOn ` X ) ) -> ( ( _I |` X ) e. ( K Cn J ) <-> J C_ K ) )
13	6, 11, 12	syl2anc 693	. . 3 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( ( _I |` X ) e. ( K Cn J ) <-> J C_ K ) )
14	5, 13	mpbird 247	. 2 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> ( _I |` X ) e. ( K Cn J ) )
15		sshauslem.3	. 2 |- ( ( J e. A /\ ( _I |` X ) : X -1-1-> X /\ ( _I |` X ) e. ( K Cn J ) ) -> K e. A )
16	1, 4, 14, 15	syl3anc 1326	1 |- ( ( J e. A /\ K e. (TopOn ` X ) /\ J C_ K ) -> K e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   _I cid 5023   |` cres 5116   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  sst0  21177  sst1  21178  sshaus  21179