Theorem cnmpt21f 20687

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
cnmpt21.j	|- ( ph -> J e. (TopOn ` X ) )
cnmpt21.k	|- ( ph -> K e. (TopOn ` Y ) )
cnmpt21.a	|- ( ph -> ( x e. X , y e. Y |-> A ) e. ( ( J tX K ) Cn L ) )
cnmpt21f.f	|- ( ph -> F e. ( L Cn M ) )

Assertion

Ref	Expression
cnmpt21f	|- ( ph -> ( x e. X , y e. Y |-> ( F ` A ) ) e. ( ( J tX K ) Cn M ) )

Distinct variable groups:   x, y, F   x, L, y   ph, x, y   x, X, y   x, M, y   x, Y, y

Allowed substitution hints:   A( x, y)   J( x, y)   K( x, y)

Proof of Theorem cnmpt21f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmpt21.j	. 2 |- ( ph -> J e. (TopOn ` X ) )
2		cnmpt21.k	. 2 |- ( ph -> K e. (TopOn ` Y ) )
3		cnmpt21.a	. 2 |- ( ph -> ( x e. X , y e. Y |-> A ) e. ( ( J tX K ) Cn L ) )
4		cnmpt21f.f	. . . 4 |- ( ph -> F e. ( L Cn M ) )
5		cntop1 20256	. . . 4 |- ( F e. ( L Cn M ) -> L e. Top )
6	4, 5	syl 17	. . 3 |- ( ph -> L e. Top )
7		eqid 2451	. . . 4 |- U. L = U. L
8	7	toptopon 19948	. . 3 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
9	6, 8	sylib 200	. 2 |- ( ph -> L e. (TopOn ` U. L ) )
10		eqid 2451	. . . . . 6 |- U. M = U. M
11	7, 10	cnf 20262	. . . . 5 |- ( F e. ( L Cn M ) -> F : U. L --> U. M )
12	4, 11	syl 17	. . . 4 |- ( ph -> F : U. L --> U. M )
13	12	feqmptd 5918	. . 3 |- ( ph -> F = ( z e. U. L |-> ( F ` z ) ) )
14	13, 4	eqeltrrd 2530	. 2 |- ( ph -> ( z e. U. L |-> ( F ` z ) ) e. ( L Cn M ) )
15		fveq2 5865	. 2 |- ( z = A -> ( F ` z ) = ( F ` A ) )
16	1, 2, 3, 9, 14, 15	cnmpt21 20686	1 |- ( ph -> ( x e. X , y e. Y |-> ( F ` A ) ) e. ( ( J tX K ) Cn M ) )

Syntax hints:   -> wi 4   e. wcel 1887   U.cuni 4198   |-> cmpt 4461   -->wf 5578   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   Topctop 19917  TopOnctopon 19918   Cn ccn 20240   tX ctx 20575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-map 7474  df-topgen 15342  df-top 19921  df-bases 19922  df-topon 19923  df-cn 20243  df-tx 20577

This theorem is referenced by:  cnmpt22  20689  cnmptk2  20701  txhmeo  20818  tgpsubcn  21105  istgp2  21106  dvrcn  21198  htpyid  22008  htpyco1  22009  reparphti  22028  pcocn  22048  pcorevlem  22057  cxpcn  23685  dipcn  26359  mndpluscn  28732  cvxscon  29966  cvmlift2lem6  30031  cvmlift2lem12  30037