Theorem icnpimaex 15093

Description: Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)

Assertion

Ref	Expression
icnpimaex	|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )

Distinct variable groups:   x, A   x, F   x, J   x, K   x, P   x, X   x, Y

Proof of Theorem icnpimaex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr3 1032	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> ( F ` P ) e. A )
2		eleq2 2298	. . . 4 |- ( y = A -> ( ( F ` P ) e. y <-> ( F ` P ) e. A ) )
3		sseq2 3264	. . . . . 6 |- ( y = A -> ( ( F " x ) C_ y <-> ( F " x ) C_ A ) )
4	3	anbi2d 464	. . . . 5 |- ( y = A -> ( ( P e. x /\ ( F " x ) C_ y ) <-> ( P e. x /\ ( F " x ) C_ A ) ) )
5	4	rexbidv 2545	. . . 4 |- ( y = A -> ( E. x e. J ( P e. x /\ ( F " x ) C_ y ) <-> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) )
6	2, 5	imbi12d 234	. . 3 |- ( y = A -> ( ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) <-> ( ( F ` P ) e. A -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) ) )
7		simpr1 1030	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> F e. ( ( J CnP K ) ` P ) )
8		iscnp 15081	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
9	8	adantr 276	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) )
10	7, 9	mpbid 147	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> ( F : X --> Y /\ A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) )
11	10	simprd 114	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> A. y e. K ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) )
12		simpr2 1031	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> A e. K )
13	6, 11, 12	rspcdva 2928	. 2 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> ( ( F ` P ) e. A -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) ) )
14	1, 13	mpd 13	1 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3213   "cima 4754   -->wf 5350   ` cfv 5354  (class class class)co 6052  TopOnctopon 14892   CnP ccnp 15068

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-top 14880  df-topon 14893  df-cnp 15071

This theorem is referenced by:  iscnp4  15100  cnpnei  15101  cnptopco  15104  cncnp  15112  cnptopresti  15120  lmtopcnp  15132  txcnp  15153