Theorem cantnfrescl 8181

Description: A function is finitely supported from B to A iff the extended function is finitely supported from D to A. (Contributed by Mario Carneiro, 25-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
cantnfrescl.d	|- ( ph -> D e. On )
cantnfrescl.b	|- ( ph -> B C_ D )
cantnfrescl.x	|- ( ( ph /\ n e. ( D \ B ) ) -> X = (/) )
cantnfrescl.a	|- ( ph -> (/) e. A )
cantnfrescl.t	|- T = dom ( A CNF D )

Assertion

Ref	Expression
cantnfrescl	|- ( ph -> ( ( n e. B |-> X ) e. S <-> ( n e. D |-> X ) e. T ) )

Distinct variable groups:   B, n   D, n   A, n   ph, n

Allowed substitution hints:   S( n)   T( n)   X( n)

Proof of Theorem cantnfrescl

Step	Hyp	Ref	Expression
1		cantnfrescl.b	. . . . 5 |- ( ph -> B C_ D )
2		cantnfrescl.x	. . . . . . 7 |- ( ( ph /\ n e. ( D \ B ) ) -> X = (/) )
3		cantnfrescl.a	. . . . . . . 8 |- ( ph -> (/) e. A )
4	3	adantr 467	. . . . . . 7 |- ( ( ph /\ n e. ( D \ B ) ) -> (/) e. A )
5	2, 4	eqeltrd 2529	. . . . . 6 |- ( ( ph /\ n e. ( D \ B ) ) -> X e. A )
6	5	ralrimiva 2802	. . . . 5 |- ( ph -> A. n e. ( D \ B ) X e. A )
7	1, 6	raldifeq 3857	. . . 4 |- ( ph -> ( A. n e. B X e. A <-> A. n e. D X e. A ) )
8		eqid 2451	. . . . 5 |- ( n e. B |-> X ) = ( n e. B |-> X )
9	8	fmpt 6043	. . . 4 |- ( A. n e. B X e. A <-> ( n e. B |-> X ) : B --> A )
10		eqid 2451	. . . . 5 |- ( n e. D |-> X ) = ( n e. D |-> X )
11	10	fmpt 6043	. . . 4 |- ( A. n e. D X e. A <-> ( n e. D |-> X ) : D --> A )
12	7, 9, 11	3bitr3g 291	. . 3 |- ( ph -> ( ( n e. B |-> X ) : B --> A <-> ( n e. D |-> X ) : D --> A ) )
13		cantnfs.b	. . . . . 6 |- ( ph -> B e. On )
14		mptexg 6135	. . . . . 6 |- ( B e. On -> ( n e. B |-> X ) e. _V )
15	13, 14	syl 17	. . . . 5 |- ( ph -> ( n e. B |-> X ) e. _V )
16		funmpt 5618	. . . . . 6 |- Fun ( n e. B |-> X )
17	16	a1i 11	. . . . 5 |- ( ph -> Fun ( n e. B |-> X ) )
18		cantnfrescl.d	. . . . . . 7 |- ( ph -> D e. On )
19		mptexg 6135	. . . . . . 7 |- ( D e. On -> ( n e. D |-> X ) e. _V )
20	18, 19	syl 17	. . . . . 6 |- ( ph -> ( n e. D |-> X ) e. _V )
21		funmpt 5618	. . . . . 6 |- Fun ( n e. D |-> X )
22	20, 21	jctir 541	. . . . 5 |- ( ph -> ( ( n e. D |-> X ) e. _V /\ Fun ( n e. D |-> X ) ) )
23	15, 17, 22	jca31 537	. . . 4 |- ( ph -> ( ( ( n e. B |-> X ) e. _V /\ Fun ( n e. B |-> X ) ) /\ ( ( n e. D |-> X ) e. _V /\ Fun ( n e. D |-> X ) ) ) )
24	18, 1, 2	extmptsuppeq 6939	. . . 4 |- ( ph -> ( ( n e. B |-> X ) supp (/) ) = ( ( n e. D |-> X ) supp (/) ) )
25		suppeqfsuppbi 7897	. . . 4 |- ( ( ( ( n e. B |-> X ) e. _V /\ Fun ( n e. B |-> X ) ) /\ ( ( n e. D |-> X ) e. _V /\ Fun ( n e. D |-> X ) ) ) -> ( ( ( n e. B |-> X ) supp (/) ) = ( ( n e. D |-> X ) supp (/) ) -> ( ( n e. B |-> X ) finSupp (/) <-> ( n e. D |-> X ) finSupp (/) ) ) )
26	23, 24, 25	sylc 62	. . 3 |- ( ph -> ( ( n e. B |-> X ) finSupp (/) <-> ( n e. D |-> X ) finSupp (/) ) )
27	12, 26	anbi12d 717	. 2 |- ( ph -> ( ( ( n e. B |-> X ) : B --> A /\ ( n e. B |-> X ) finSupp (/) ) <-> ( ( n e. D |-> X ) : D --> A /\ ( n e. D |-> X ) finSupp (/) ) ) )
28		cantnfs.s	. . 3 |- S = dom ( A CNF B )
29		cantnfs.a	. . 3 |- ( ph -> A e. On )
30	28, 29, 13	cantnfs 8171	. 2 |- ( ph -> ( ( n e. B |-> X ) e. S <-> ( ( n e. B |-> X ) : B --> A /\ ( n e. B |-> X ) finSupp (/) ) ) )
31		cantnfrescl.t	. . 3 |- T = dom ( A CNF D )
32	31, 29, 18	cantnfs 8171	. 2 |- ( ph -> ( ( n e. D |-> X ) e. T <-> ( ( n e. D |-> X ) : D --> A /\ ( n e. D |-> X ) finSupp (/) ) ) )
33	27, 30, 32	3bitr4d 289	1 |- ( ph -> ( ( n e. B |-> X ) e. S <-> ( n e. D |-> X ) e. T ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   A.wral 2737   _Vcvv 3045   \ cdif 3401   C_ wss 3404   (/)c0 3731   class class class wbr 4402   |-> cmpt 4461   dom cdm 4834   Oncon0 5423   Fun wfun 5576   -->wf 5578  (class class class)co 6290   supp csupp 6914   finSupp cfsupp 7883   CNF ccnf 8166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-seqom 7165  df-map 7474  df-fsupp 7884  df-cnf 8167

This theorem is referenced by:  cantnfres  8182