Theorem cnmpt11f 14756

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 14674	. . . 4 |- ( ( x e. X |-> A ) e. ( J Cn K ) -> K e. Top )
4	2, 3	syl 14	. . 3 |- ( ph -> K e. Top )
5		eqid 2205	. . . 4 |- U. K = U. K
6	5	toptopon 14490	. . 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 2205	. . . . . 6 |- U. L = U. L
10	5, 9	cnf 14676	. . . . 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 5632	. . 3 |- ( ph -> F = ( y e. U. K |-> ( F ` y ) ) )
13	12, 8	eqeltrrd 2283	. 2 |- ( ph -> ( y e. U. K |-> ( F ` y ) ) e. ( K Cn L ) )
14		fveq2 5576	. 2 |- ( y = A -> ( F ` y ) = ( F ` A ) )
15	1, 2, 7, 13, 14	cnmpt11 14755	1 |- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( J Cn L ) )

Syntax hints:   -> wi 4   e. wcel 2176   U.cuni 3850   |-> cmpt 4105   -->wf 5267   ` cfv 5271  (class class class)co 5944   Topctop 14469  TopOnctopon 14482   Cn ccn 14657

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-top 14470  df-topon 14483  df-cn 14660

This theorem is referenced by:  cnmpt12f  14758