Theorem cnmpt1k 21485

Description: The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmptk1.j	|- ( ph -> J e. (TopOn ` X ) )
cnmptk1.k	|- ( ph -> K e. (TopOn ` Y ) )
cnmptk1.l	|- ( ph -> L e. (TopOn ` Z ) )
cnmpt1k.m	|- ( ph -> M e. (TopOn ` W ) )
cnmpt1k.a	|- ( ph -> ( x e. X |-> A ) e. ( J Cn L ) )
cnmpt1k.b	|- ( ph -> ( y e. Y |-> ( z e. Z |-> B ) ) e. ( K Cn ( M ^ko L ) ) )
cnmpt1k.c	|- ( z = A -> B = C )

Assertion

Ref	Expression
cnmpt1k	|- ( ph -> ( y e. Y |-> ( x e. X |-> C ) ) e. ( K Cn ( M ^ko J ) ) )

Distinct variable groups:   x, y, J   x, K, y   x, L, y   x, M, y   x, z, Z, y   z, A   x, B   ph, x, y   x, X, y   x, Y, y   z, C   y, A

Allowed substitution hints:   ph( z)   A( x)   B( y, z)   C( x, y)   J( z)   K( z)   L( z)   M( z)   W( x, y, z)   X( z)   Y( z)

Proof of Theorem cnmpt1k

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnmptk1.j	. . . . . . 7 |- ( ph -> J e. (TopOn ` X ) )
2		cnmptk1.l	. . . . . . 7 |- ( ph -> L e. (TopOn ` Z ) )
3		cnmpt1k.a	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( J Cn L ) )
4		cnf2 21053	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ L e. (TopOn ` Z ) /\ ( x e. X |-> A ) e. ( J Cn L ) ) -> ( x e. X |-> A ) : X --> Z )
5	1, 2, 3, 4	syl3anc 1326	. . . . . 6 |- ( ph -> ( x e. X |-> A ) : X --> Z )
6		eqid 2622	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
7	6	fmpt 6381	. . . . . 6 |- ( A. x e. X A e. Z <-> ( x e. X |-> A ) : X --> Z )
8	5, 7	sylibr 224	. . . . 5 |- ( ph -> A. x e. X A e. Z )
9	8	adantr 481	. . . 4 |- ( ( ph /\ y e. Y ) -> A. x e. X A e. Z )
10		eqidd 2623	. . . 4 |- ( ( ph /\ y e. Y ) -> ( x e. X |-> A ) = ( x e. X |-> A ) )
11		eqidd 2623	. . . 4 |- ( ( ph /\ y e. Y ) -> ( z e. Z |-> B ) = ( z e. Z |-> B ) )
12		cnmpt1k.c	. . . 4 |- ( z = A -> B = C )
13	9, 10, 11, 12	fmptcof 6397	. . 3 |- ( ( ph /\ y e. Y ) -> ( ( z e. Z |-> B ) o. ( x e. X |-> A ) ) = ( x e. X |-> C ) )
14	13	mpteq2dva 4744	. 2 |- ( ph -> ( y e. Y |-> ( ( z e. Z |-> B ) o. ( x e. X |-> A ) ) ) = ( y e. Y |-> ( x e. X |-> C ) ) )
15		cnmptk1.k	. . 3 |- ( ph -> K e. (TopOn ` Y ) )
16		cnmpt1k.b	. . 3 |- ( ph -> ( y e. Y |-> ( z e. Z |-> B ) ) e. ( K Cn ( M ^ko L ) ) )
17		topontop 20718	. . . . 5 |- ( L e. (TopOn ` Z ) -> L e. Top )
18	2, 17	syl 17	. . . 4 |- ( ph -> L e. Top )
19		cnmpt1k.m	. . . . 5 |- ( ph -> M e. (TopOn ` W ) )
20		topontop 20718	. . . . 5 |- ( M e. (TopOn ` W ) -> M e. Top )
21	19, 20	syl 17	. . . 4 |- ( ph -> M e. Top )
22		eqid 2622	. . . . 5 |- ( M ^ko L ) = ( M ^ko L )
23	22	xkotopon 21403	. . . 4 |- ( ( L e. Top /\ M e. Top ) -> ( M ^ko L ) e. (TopOn ` ( L Cn M ) ) )
24	18, 21, 23	syl2anc 693	. . 3 |- ( ph -> ( M ^ko L ) e. (TopOn ` ( L Cn M ) ) )
25	21, 3	xkoco1cn 21460	. . 3 |- ( ph -> ( w e. ( L Cn M ) |-> ( w o. ( x e. X |-> A ) ) ) e. ( ( M ^ko L ) Cn ( M ^ko J ) ) )
26		coeq1 5279	. . 3 |- ( w = ( z e. Z |-> B ) -> ( w o. ( x e. X |-> A ) ) = ( ( z e. Z |-> B ) o. ( x e. X |-> A ) ) )
27	15, 16, 24, 25, 26	cnmpt11 21466	. 2 |- ( ph -> ( y e. Y |-> ( ( z e. Z |-> B ) o. ( x e. X |-> A ) ) ) e. ( K Cn ( M ^ko J ) ) )
28	14, 27	eqeltrrd 2702	1 |- ( ph -> ( y e. Y |-> ( x e. X |-> C ) ) e. ( K Cn ( M ^ko J ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ^ko cxko 21364

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-rep 4771  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-3or 1038  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cmp 21190  df-xko 21366

This theorem is referenced by: (None)