Theorem cnmpt11f 14168

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 14086	. . . 4 |- ( ( x e. X |-> A ) e. ( J Cn K ) -> K e. Top )
4	2, 3	syl 14	. . 3 |- ( ph -> K e. Top )
5		eqid 2189	. . . 4 |- U. K = U. K
6	5	toptopon 13902	. . 3 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
7	4, 6	sylib 122	. 2 |- ( ph -> K e. (TopOn ` U. K ) )
8		cnmpt11f.f	. . . . 5 |- ( ph -> F e. ( K Cn L ) )
9		eqid 2189	. . . . . 6 |- U. L = U. L
10	5, 9	cnf 14088	. . . . 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 5585	. . 3 |- ( ph -> F = ( y e. U. K |-> ( F ` y ) ) )
13	12, 8	eqeltrrd 2267	. 2 |- ( ph -> ( y e. U. K |-> ( F ` y ) ) e. ( K Cn L ) )
14		fveq2 5530	. 2 |- ( y = A -> ( F ` y ) = ( F ` A ) )
15	1, 2, 7, 13, 14	cnmpt11 14167	1 |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Syntax hints:   -> wi 4   e. wcel 2160   U.cuni 3824   |-> cmpt 4079   -->wf 5227   ` cfv 5231  (class class class)co 5891   Topctop 13881  TopOnctopon 13894   Cn ccn 14069

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-map 6668  df-top 13882  df-topon 13895  df-cn 14072

This theorem is referenced by:  cnmpt12f  14170