Theorem cnmpt22f 21478

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 21045	. . . 4 |- ( ( x e. X , y e. Y |-> A ) e. ( ( J tX K ) Cn L ) -> L e. Top )
6	3, 5	syl 17	. . 3 |- ( ph -> L e. Top )
7		eqid 2622	. . . 4 |- U. L = U. L
8	7	toptopon 20722	. . 3 |- ( L e. Top <-> L e. (TopOn ` U. L ) )
9	6, 8	sylib 208	. 2 |- ( ph -> L e. (TopOn ` U. L ) )
10		cntop2 21045	. . . 4 |- ( ( x e. X , y e. Y |-> B ) e. ( ( J tX K ) Cn M ) -> M e. Top )
11	4, 10	syl 17	. . 3 |- ( ph -> M e. Top )
12		eqid 2622	. . . 4 |- U. M = U. M
13	12	toptopon 20722	. . 3 |- ( M e. Top <-> M e. (TopOn ` U. M ) )
14	11, 13	sylib 208	. 2 |- ( ph -> M e. (TopOn ` U. M ) )
15		txtopon 21394	. . . . . . 7 |- ( ( L e. (TopOn ` U. L ) /\ M e. (TopOn ` U. M ) ) -> ( L tX M ) e. (TopOn ` ( U. L X. U. M ) ) )
16	9, 14, 15	syl2anc 693	. . . . . 6 |- ( ph -> ( L tX M ) e. (TopOn ` ( U. L X. U. M ) ) )
17		cnmpt22f.f	. . . . . . . 8 |- ( ph -> F e. ( ( L tX M ) Cn N ) )
18		cntop2 21045	. . . . . . . 8 |- ( F e. ( ( L tX M ) Cn N ) -> N e. Top )
19	17, 18	syl 17	. . . . . . 7 |- ( ph -> N e. Top )
20		eqid 2622	. . . . . . . 8 |- U. N = U. N
21	20	toptopon 20722	. . . . . . 7 |- ( N e. Top <-> N e. (TopOn ` U. N ) )
22	19, 21	sylib 208	. . . . . 6 |- ( ph -> N e. (TopOn ` U. N ) )
23		cnf2 21053	. . . . . 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 )
24	16, 22, 17, 23	syl3anc 1326	. . . . 5 |- ( ph -> F : ( U. L X. U. M ) --> U. N )
25		ffn 6045	. . . . 5 |- ( F : ( U. L X. U. M ) --> U. N -> F Fn ( U. L X. U. M ) )
26	24, 25	syl 17	. . . 4 |- ( ph -> F Fn ( U. L X. U. M ) )
27		fnov 6768	. . . 4 |- ( F Fn ( U. L X. U. M ) <-> F = ( z e. U. L , w e. U. M |-> ( z F w ) ) )
28	26, 27	sylib 208	. . 3 |- ( ph -> F = ( z e. U. L , w e. U. M |-> ( z F w ) ) )
29	28, 17	eqeltrrd 2702	. 2 |- ( ph -> ( z e. U. L , w e. U. M |-> ( z F w ) ) e. ( ( L tX M ) Cn N ) )
30		oveq12 6659	. 2 |- ( ( z = A /\ w = B ) -> ( z F w ) = ( A F B ) )
31	1, 2, 3, 4, 9, 14, 29, 30	cnmpt22 21477	1 |- ( ph -> ( x e. X , y e. Y |-> ( A F B ) ) e. ( ( J tX K ) Cn N ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-tx 21365

This theorem is referenced by:  cnmptcom  21481  cnmpt2plusg  21892  istgp2  21895  cnmpt2vsca  21998  cnmpt2ds  22646  divcn  22671  cnrehmeo  22752  htpycom  22775  htpyco1  22777  htpycc  22779  reparphti  22797  pcohtpylem  22819  cnmpt2ip  23047  cxpcn  24486  vmcn  27554  dipcn  27575  mndpluscn  29972  cvxsconn  31225