Theorem cnmpt22f 15106

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 ) )
cnmpt2t.b	|- ( ph -> ( x e. X , y e. Y |-> B ) e. ( ( J tX K ) Cn M ) )
cnmpt22f.f	|- ( ph -> F e. ( ( L tX M ) Cn N ) )

Assertion

Ref	Expression
cnmpt22f	|- ( ph -> ( x e. X , y e. Y |-> ( A F B ) ) e. ( ( J tX K ) Cn N ) )

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

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

Proof of Theorem cnmpt22f

Dummy variables w z are mutually distinct and 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		cnmpt2t.b	. 2 |- ( ph -> ( x e. X , y e. Y |-> B ) e. ( ( J tX K ) Cn M ) )
5		cntop2 15013	. . . 4 |- ( ( x e. X , y e. Y |-> A ) e. ( ( J tX K ) Cn L ) -> L e. Top )
6	3, 5	syl 14	. . 3 |- ( ph -> L e. Top )
7		toptopon2 14830	. . 3 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
8	6, 7	sylib 122	. 2 |- ( ph -> L e. (TopOn ` U. L ) )
9		cntop2 15013	. . . 4 |- ( ( x e. X , y e. Y |-> B ) e. ( ( J tX K ) Cn M ) -> M e. Top )
10	4, 9	syl 14	. . 3 |- ( ph -> M e. Top )
11		toptopon2 14830	. . 3 |- ( M e. Top <-> M e. (TopOn ` U. M ) )
12	10, 11	sylib 122	. 2 |- ( ph -> M e. (TopOn ` U. M ) )
13		txtopon 15073	. . . . . . 7 |- ( ( L e. (TopOn ` U. L ) /\ M e. (TopOn ` U. M ) ) -> ( L tX M ) e. (TopOn ` ( U. L X. U. M ) ) )
14	8, 12, 13	syl2anc 411	. . . . . 6 |- ( ph -> ( L tX M ) e. (TopOn ` ( U. L X. U. M ) ) )
15		cnmpt22f.f	. . . . . . . 8 |- ( ph -> F e. ( ( L tX M ) Cn N ) )
16		cntop2 15013	. . . . . . . 8 |- ( F e. ( ( L tX M ) Cn N ) -> N e. Top )
17	15, 16	syl 14	. . . . . . 7 |- ( ph -> N e. Top )
18		toptopon2 14830	. . . . . . 7 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
19	17, 18	sylib 122	. . . . . 6 |- ( ph -> N e. (TopOn ` U. N ) )
20		cnf2 15016	. . . . . 6 |- ( ( ( L tX M ) e. (TopOn ` ( U. L X. U. M ) ) /\ N e. (TopOn ` U. N ) /\ F e. ( ( L tX M ) Cn N ) ) -> F : ( U. L X. U. M ) --> U. N )
21	14, 19, 15, 20	syl3anc 1274	. . . . 5 |- ( ph -> F : ( U. L X. U. M ) --> U. N )
22	21	ffnd 5490	. . . 4 |- ( ph -> F Fn ( U. L X. U. M ) )
23		fnovim 6140	. . . 4 |- ( F Fn ( U. L X. U. M ) -> F = ( z e. U. L , w e. U. M |-> ( z F w ) ) )
24	22, 23	syl 14	. . 3 |- ( ph -> F = ( z e. U. L , w e. U. M |-> ( z F w ) ) )
25	24, 15	eqeltrrd 2309	. 2 |- ( ph -> ( z e. U. L , w e. U. M |-> ( z F w ) ) e. ( ( L tX M ) Cn N ) )
26		oveq12 6037	. 2 |- ( ( z = A /\ w = B ) -> ( z F w ) = ( A F B ) )
27	1, 2, 3, 4, 8, 12, 25, 26	cnmpt22 15105	1 |- ( ph -> ( x e. X , y e. Y |-> ( A F B ) ) e. ( ( J tX K ) Cn N ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   U.cuni 3898   X. cxp 4729   Fn wfn 5328   -->wf 5329   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   Topctop 14808  TopOnctopon 14821   Cn ccn 14996   tX ctx 15063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-topgen 13423  df-top 14809  df-topon 14822  df-bases 14854  df-cn 14999  df-tx 15064

This theorem is referenced by:  cnmptcom  15109  divcnap  15376  cnrehmeocntop  15421