Theorem cnmpt11f 12924

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 12842	. . . 4 |- ( ( x e. X |-> A ) e. ( J Cn K ) -> K e. Top )
4	2, 3	syl 14	. . 3 |- ( ph -> K e. Top )
5		eqid 2165	. . . 4 |- U. K = U. K
6	5	toptopon 12656	. . 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 2165	. . . . . 6 |- U. L = U. L
10	5, 9	cnf 12844	. . . . 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 5539	. . 3 |- ( ph -> F = ( y e. U. K |-> ( F ` y ) ) )
13	12, 8	eqeltrrd 2244	. 2 |- ( ph -> ( y e. U. K |-> ( F ` y ) ) e. ( K Cn L ) )
14		fveq2 5486	. 2 |- ( y = A -> ( F ` y ) = ( F ` A ) )
15	1, 2, 7, 13, 14	cnmpt11 12923	1 |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Syntax hints:   -> wi 4   e. wcel 2136   U.cuni 3789   |-> cmpt 4043   -->wf 5184   ` cfv 5188  (class class class)co 5842   Topctop 12635  TopOnctopon 12648   Cn ccn 12825

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-top 12636  df-topon 12649  df-cn 12828

This theorem is referenced by:  cnmpt12f  12926