Theorem cnmpt11f 12442

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptid.j	|- ( ph -> J e. (TopOn ` X ) )
cnmpt11.a	|- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )
cnmpt11f.f	|- ( ph -> F e. ( K Cn L ) )

Assertion

Ref	Expression
cnmpt11f	|- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Distinct variable groups:   x, F   ph, x   x, J   x, X   x, K   x, L

Allowed substitution hint:   A( x)

Proof of Theorem cnmpt11f

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptid.j	. 2 |- ( ph -> J e. (TopOn ` X ) )
2		cnmpt11.a	. 2 |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )
3		cntop2 12360	. . . 4 |- ( ( x e. X |-> A ) e. ( J Cn K ) -> K e. Top )
4	2, 3	syl 14	. . 3 |- ( ph -> K e. Top )
5		eqid 2137	. . . 4 |- U. K = U. K
6	5	toptopon 12174	. . 3 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
7	4, 6	sylib 121	. 2 |- ( ph -> K e. (TopOn ` U. K ) )
8		cnmpt11f.f	. . . . 5 |- ( ph -> F e. ( K Cn L ) )
9		eqid 2137	. . . . . 6 |- U. L = U. L
10	5, 9	cnf 12362	. . . . 5 |- ( F e. ( K Cn L ) -> F : U. K --> U. L )
11	8, 10	syl 14	. . . 4 |- ( ph -> F : U. K --> U. L )
12	11	feqmptd 5467	. . 3 |- ( ph -> F = ( y e. U. K |-> ( F ` y ) ) )
13	12, 8	eqeltrrd 2215	. 2 |- ( ph -> ( y e. U. K |-> ( F ` y ) ) e. ( K Cn L ) )
14		fveq2 5414	. 2 |- ( y = A -> ( F ` y ) = ( F ` A ) )
15	1, 2, 7, 13, 14	cnmpt11 12441	1 |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Syntax hints:   -> wi 4   e. wcel 1480   U.cuni 3731   |-> cmpt 3984   -->wf 5114   ` cfv 5118  (class class class)co 5767   Topctop 12153  TopOnctopon 12166   Cn ccn 12343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-top 12154  df-topon 12167  df-cn 12346

This theorem is referenced by:  cnmpt12f  12444