Theorem kqfvima 21533

Description: When the image set is open, the quotient map satisfies a partial converse to fnfvima 6496, which is normally only true for injective functions. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	|- F = ( x e. X |-> { y e. J | x e. y } )

Assertion

Ref	Expression
kqfvima	|- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> ( A e. U <-> ( F ` A ) e. ( F " U ) ) )

Distinct variable groups:   x, y, A   x, J, y   x, X, y

Allowed substitution hints:   U( x, y)   F( x, y)

Proof of Theorem kqfvima

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . 5 |- F = ( x e. X |-> { y e. J | x e. y } )
2	1	kqffn 21528	. . . 4 |- ( J e. (TopOn ` X ) -> F Fn X )
3	2	3ad2ant1 1082	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> F Fn X )
4		toponss 20731	. . . 4 |- ( ( J e. (TopOn ` X ) /\ U e. J ) -> U C_ X )
5	4	3adant3 1081	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> U C_ X )
6		fnfvima 6496	. . . 4 |- ( ( F Fn X /\ U C_ X /\ A e. U ) -> ( F ` A ) e. ( F " U ) )
7	6	3expia 1267	. . 3 |- ( ( F Fn X /\ U C_ X ) -> ( A e. U -> ( F ` A ) e. ( F " U ) ) )
8	3, 5, 7	syl2anc 693	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> ( A e. U -> ( F ` A ) e. ( F " U ) ) )
9		fnfun 5988	. . . 4 |- ( F Fn X -> Fun F )
10		fvelima 6248	. . . . 5 |- ( ( Fun F /\ ( F ` A ) e. ( F " U ) ) -> E. z e. U ( F ` z ) = ( F ` A ) )
11	10	ex 450	. . . 4 |- ( Fun F -> ( ( F ` A ) e. ( F " U ) -> E. z e. U ( F ` z ) = ( F ` A ) ) )
12	3, 9, 11	3syl 18	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> ( ( F ` A ) e. ( F " U ) -> E. z e. U ( F ` z ) = ( F ` A ) ) )
13		simpl1 1064	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> J e. (TopOn ` X ) )
14	5	sselda 3603	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> z e. X )
15		simpl3 1066	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> A e. X )
16	1	kqfeq 21527	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ z e. X /\ A e. X ) -> ( ( F ` z ) = ( F ` A ) <-> A. y e. J ( z e. y <-> A e. y ) ) )
17	13, 14, 15, 16	syl3anc 1326	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> ( ( F ` z ) = ( F ` A ) <-> A. y e. J ( z e. y <-> A e. y ) ) )
18		eleq2 2690	. . . . . . . . 9 |- ( y = w -> ( z e. y <-> z e. w ) )
19		eleq2 2690	. . . . . . . . 9 |- ( y = w -> ( A e. y <-> A e. w ) )
20	18, 19	bibi12d 335	. . . . . . . 8 |- ( y = w -> ( ( z e. y <-> A e. y ) <-> ( z e. w <-> A e. w ) ) )
21	20	cbvralv 3171	. . . . . . 7 |- ( A. y e. J ( z e. y <-> A e. y ) <-> A. w e. J ( z e. w <-> A e. w ) )
22	17, 21	syl6bb 276	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> ( ( F ` z ) = ( F ` A ) <-> A. w e. J ( z e. w <-> A e. w ) ) )
23		simpl2 1065	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> U e. J )
24		eleq2 2690	. . . . . . . . 9 |- ( w = U -> ( z e. w <-> z e. U ) )
25		eleq2 2690	. . . . . . . . 9 |- ( w = U -> ( A e. w <-> A e. U ) )
26	24, 25	bibi12d 335	. . . . . . . 8 |- ( w = U -> ( ( z e. w <-> A e. w ) <-> ( z e. U <-> A e. U ) ) )
27	26	rspcv 3305	. . . . . . 7 |- ( U e. J -> ( A. w e. J ( z e. w <-> A e. w ) -> ( z e. U <-> A e. U ) ) )
28	23, 27	syl 17	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> ( A. w e. J ( z e. w <-> A e. w ) -> ( z e. U <-> A e. U ) ) )
29	22, 28	sylbid 230	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> ( ( F ` z ) = ( F ` A ) -> ( z e. U <-> A e. U ) ) )
30		simpr 477	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> z e. U )
31		biimp 205	. . . . 5 |- ( ( z e. U <-> A e. U ) -> ( z e. U -> A e. U ) )
32	29, 30, 31	syl6ci 71	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) /\ z e. U ) -> ( ( F ` z ) = ( F ` A ) -> A e. U ) )
33	32	rexlimdva 3031	. . 3 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> ( E. z e. U ( F ` z ) = ( F ` A ) -> A e. U ) )
34	12, 33	syld 47	. 2 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> ( ( F ` A ) e. ( F " U ) -> A e. U ) )
35	8, 34	impbid 202	1 |- ( ( J e. (TopOn ` X ) /\ U e. J /\ A e. X ) -> ( A e. U <-> ( F ` A ) e. ( F " U ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   |-> cmpt 4729   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  TopOnctopon 20715

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-fv 5896  df-topon 20716

This theorem is referenced by:  kqsat  21534  kqdisj  21535  kqcldsat  21536  kqt0lem  21539  isr0  21540  regr1lem  21542  kqreglem1  21544  kqreglem2  21545