Theorem cnmpt22f 13880

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 13787	. . . 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 13604	. . 3 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
8	6, 7	sylib 122	. 2 |- ( ph -> L e. (TopOn ` U. L ) )
9		cntop2 13787	. . . 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 13604	. . 3 |- ( M e. Top <-> M e. (TopOn ` U. M ) )
12	10, 11	sylib 122	. 2 |- ( ph -> M e. (TopOn ` U. M ) )
13		txtopon 13847	. . . . . . 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 13787	. . . . . . . 8 |- ( F e. ( ( L tX M ) Cn N ) -> N e. Top )
17	15, 16	syl 14	. . . . . . 7 |- ( ph -> N e. Top )
18		toptopon2 13604	. . . . . . 7 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
19	17, 18	sylib 122	. . . . . 6 |- ( ph -> N e. (TopOn ` U. N ) )
20		cnf2 13790	. . . . . 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 1238	. . . . 5 |- ( ph -> F : ( U. L X. U. M ) --> U. N )
22	21	ffnd 5368	. . . 4 |- ( ph -> F Fn ( U. L X. U. M ) )
23		fnovim 5985	. . . 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 2255	. 2 |- ( ph -> ( z e. U. L , w e. U. M |-> ( z F w ) ) e. ( ( L tX M ) Cn N ) )
26		oveq12 5886	. 2 |- ( ( z = A /\ w = B ) -> ( z F w ) = ( A F B ) )
27	1, 2, 3, 4, 8, 12, 25, 26	cnmpt22 13879	1 |- ( ph -> ( x e. X , y e. Y |-> ( A F B ) ) e. ( ( J tX K ) Cn N ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   U.cuni 3811   X. cxp 4626   Fn wfn 5213   -->wf 5214   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   Topctop 13582  TopOnctopon 13595   Cn ccn 13770   tX ctx 13837

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-topgen 12714  df-top 13583  df-topon 13596  df-bases 13628  df-cn 13773  df-tx 13838

This theorem is referenced by:  cnmptcom  13883  divcnap  14140  cnrehmeocntop  14178