Theorem cnmpt22f 14882

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 14789	. . . 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 14606	. . 3 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
8	6, 7	sylib 122	. 2 |- ( ph -> L e. (TopOn ` U. L ) )
9		cntop2 14789	. . . 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 14606	. . 3 |- ( M e. Top <-> M e. (TopOn ` U. M ) )
12	10, 11	sylib 122	. 2 |- ( ph -> M e. (TopOn ` U. M ) )
13		txtopon 14849	. . . . . . 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 14789	. . . . . . . 8 |- ( F e. ( ( L tX M ) Cn N ) -> N e. Top )
17	15, 16	syl 14	. . . . . . 7 |- ( ph -> N e. Top )
18		toptopon2 14606	. . . . . . 7 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
19	17, 18	sylib 122	. . . . . 6 |- ( ph -> N e. (TopOn ` U. N ) )
20		cnf2 14792	. . . . . 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 1250	. . . . 5 |- ( ph -> F : ( U. L X. U. M ) --> U. N )
22	21	ffnd 5446	. . . 4 |- ( ph -> F Fn ( U. L X. U. M ) )
23		fnovim 6077	. . . 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 2285	. 2 |- ( ph -> ( z e. U. L , w e. U. M |-> ( z F w ) ) e. ( ( L tX M ) Cn N ) )
26		oveq12 5976	. 2 |- ( ( z = A /\ w = B ) -> ( z F w ) = ( A F B ) )
27	1, 2, 3, 4, 8, 12, 25, 26	cnmpt22 14881	1 |- ( ph -> ( x e. X , y e. Y |-> ( A F B ) ) e. ( ( J tX K ) Cn N ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   U.cuni 3864   X. cxp 4691   Fn wfn 5285   -->wf 5286   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   Topctop 14584  TopOnctopon 14597   Cn ccn 14772   tX ctx 14839

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-topgen 13207  df-top 14585  df-topon 14598  df-bases 14630  df-cn 14775  df-tx 14840

This theorem is referenced by:  cnmptcom  14885  divcnap  15152  cnrehmeocntop  15197