Theorem cnmpt12f 15277

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
cnmptid.j	|- ( ph -> J e. (TopOn ` X ) )
cnmpt11.a	|- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )
cnmpt1t.b	|- ( ph -> ( x e. X |-> B ) e. ( J Cn L ) )
cnmpt12f.f	|- ( ph -> F e. ( ( K tX L ) Cn M ) )

Assertion

Ref	Expression
cnmpt12f	|- ( ph -> ( x e. X |-> ( A F B ) ) e. ( J Cn M ) )

Distinct variable groups:   x, F   ph, x   x, J   x, M   x, X   x, K   x, L

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem cnmpt12f

Step	Hyp	Ref	Expression
1		df-ov 6061	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	mpteq2i 4202	. 2 |- ( x e. X |-> ( A F B ) ) = ( x e. X |-> ( F ` <. A , B >. ) )
3		cnmptid.j	. . 3 |- ( ph -> J e. (TopOn ` X ) )
4		cnmpt11.a	. . . 4 |- ( ph -> ( x e. X |-> A ) e. ( J Cn K ) )
5		cnmpt1t.b	. . . 4 |- ( ph -> ( x e. X |-> B ) e. ( J Cn L ) )
6	3, 4, 5	cnmpt1t 15276	. . 3 |- ( ph -> ( x e. X |-> <. A , B >. ) e. ( J Cn ( K tX L ) ) )
7		cnmpt12f.f	. . 3 |- ( ph -> F e. ( ( K tX L ) Cn M ) )
8	3, 6, 7	cnmpt11f 15275	. 2 |- ( ph -> ( x e. X |-> ( F ` <. A , B >. ) ) e. ( J Cn M ) )
9	2, 8	eqeltrid 2321	1 |- ( ph -> ( x e. X |-> ( A F B ) ) e. ( J Cn M ) )

Syntax hints:   -> wi 4   e. wcel 2205   <.cop 3697   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058  TopOnctopon 15001   Cn ccn 15176   tX ctx 15243

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-topgen 13557  df-top 14989  df-topon 15002  df-bases 15034  df-cn 15179  df-tx 15244

This theorem is referenced by:  cnmpt12  15278  fsumcncntop  15558  expcn  15560  cncfmpt2fcntop  15590