Theorem icnpimaex 14447

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 1007	. 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 2260	. . . 4 |- ( y = A -> ( ( F ` P ) e. y <-> ( F ` P ) e. A ) )
3		sseq2 3207	. . . . . 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 2498	. . . 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 1005	. . . . 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 14435	. . . . . 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 1006	. . 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 2873	. 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 980   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   "cima 4666   -->wf 5254   ` cfv 5258  (class class class)co 5922  TopOnctopon 14246   CnP ccnp 14422

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-top 14234  df-topon 14247  df-cnp 14425

This theorem is referenced by:  iscnp4  14454  cnpnei  14455  cnptopco  14458  cncnp  14466  cnptopresti  14474  lmtopcnp  14486  txcnp  14507