Theorem cnco 15245

Description: The composition of two continuous functions is a continuous function. (Contributed by FL, 8-Dec-2006.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
cnco	|- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) e. ( J Cn L ) )

Proof of Theorem cnco

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntop1 15225	. . 3 |- ( F e. ( J Cn K ) -> J e. Top )
2		cntop2 15226	. . 3 |- ( G e. ( K Cn L ) -> L e. Top )
3	1, 2	anim12i 338	. 2 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( J e. Top /\ L e. Top ) )
4		eqid 2238	. . . . 5 |- U. K = U. K
5		eqid 2238	. . . . 5 |- U. L = U. L
6	4, 5	cnf 15228	. . . 4 |- ( G e. ( K Cn L ) -> G : U. K --> U. L )
7		eqid 2238	. . . . 5 |- U. J = U. J
8	7, 4	cnf 15228	. . . 4 |- ( F e. ( J Cn K ) -> F : U. J --> U. K )
9		fco 5547	. . . 4 |- ( ( G : U. K --> U. L /\ F : U. J --> U. K ) -> ( G o. F ) : U. J --> U. L )
10	6, 8, 9	syl2anr 290	. . 3 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) : U. J --> U. L )
11		cnvco 4960	. . . . . . 7 |- `' ( G o. F ) = ( `' F o. `' G )
12	11	imaeq1i 5118	. . . . . 6 |- ( `' ( G o. F ) " x ) = ( ( `' F o. `' G ) " x )
13		imaco 5288	. . . . . 6 |- ( ( `' F o. `' G ) " x ) = ( `' F " ( `' G " x ) )
14	12, 13	eqtri 2259	. . . . 5 |- ( `' ( G o. F ) " x ) = ( `' F " ( `' G " x ) )
15		simpll 531	. . . . . 6 |- ( ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) /\ x e. L ) -> F e. ( J Cn K ) )
16		cnima 15244	. . . . . . 7 |- ( ( G e. ( K Cn L ) /\ x e. L ) -> ( `' G " x ) e. K )
17	16	adantll 480	. . . . . 6 |- ( ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) /\ x e. L ) -> ( `' G " x ) e. K )
18		cnima 15244	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ ( `' G " x ) e. K ) -> ( `' F " ( `' G " x ) ) e. J )
19	15, 17, 18	syl2anc 415	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) /\ x e. L ) -> ( `' F " ( `' G " x ) ) e. J )
20	14, 19	eqeltrid 2325	. . . 4 |- ( ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) /\ x e. L ) -> ( `' ( G o. F ) " x ) e. J )
21	20	ralrimiva 2623	. . 3 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> A. x e. L ( `' ( G o. F ) " x ) e. J )
22	10, 21	jca 306	. 2 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( ( G o. F ) : U. J --> U. L /\ A. x e. L ( `' ( G o. F ) " x ) e. J ) )
23	7, 5	iscn2 15224	. 2 |- ( ( G o. F ) e. ( J Cn L ) <-> ( ( J e. Top /\ L e. Top ) /\ ( ( G o. F ) : U. J --> U. L /\ A. x e. L ( `' ( G o. F ) " x ) e. J ) ) )
24	3, 22, 23	sylanbrc 421	1 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) e. ( J Cn L ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528   U.cuni 3930   `'ccnv 4768   "cima 4772   o. ccom 4773   -->wf 5368  (class class class)co 6075   Topctop 15021   Cn ccn 15209

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-top 15022  df-topon 15035  df-cn 15212

This theorem is referenced by:  txcn  15299  cnmpt11  15307  cnmpt21  15315  hmeoco  15340