Theorem cantnfrescl 8573

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 481	. . . . . . 7 |- ( ( ph /\ n e. ( D \ B ) ) -> (/) e. A )
5	2, 4	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ n e. ( D \ B ) ) -> X e. A )
6	5	ralrimiva 2966	. . . . 5 |- ( ph -> A. n e. ( D \ B ) X e. A )
7	1, 6	raldifeq 4059	. . . 4 |- ( ph -> ( A. n e. B X e. A <-> A. n e. D X e. A ) )
8		eqid 2622	. . . . 5 |- ( n e. B |-> X ) = ( n e. B |-> X )
9	8	fmpt 6381	. . . 4 |- ( A. n e. B X e. A <-> ( n e. B |-> X ) : B --> A )
10		eqid 2622	. . . . 5 |- ( n e. D |-> X ) = ( n e. D |-> X )
11	10	fmpt 6381	. . . 4 |- ( A. n e. D X e. A <-> ( n e. D |-> X ) : D --> A )
12	7, 9, 11	3bitr3g 302	. . 3 |- ( ph -> ( ( n e. B |-> X ) : B --> A <-> ( n e. D |-> X ) : D --> A ) )
13		cantnfs.b	. . . . . 6 |- ( ph -> B e. On )
14		mptexg 6484	. . . . . 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 5926	. . . . . 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 6484	. . . . . . 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 5926	. . . . . 6 |- Fun ( n e. D |-> X )
22	20, 21	jctir 561	. . . . 5 |- ( ph -> ( ( n e. D |-> X ) e. _V /\ Fun ( n e. D |-> X ) ) )
23	15, 17, 22	jca31 557	. . . 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 7319	. . . 4 |- ( ph -> ( ( n e. B |-> X ) supp (/) ) = ( ( n e. D |-> X ) supp (/) ) )
25		suppeqfsuppbi 8289	. . . 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 65	. . 3 |- ( ph -> ( ( n e. B |-> X ) finSupp (/) <-> ( n e. D |-> X ) finSupp (/) ) )
27	12, 26	anbi12d 747	. 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 8563	. 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 8563	. 2 |- ( ph -> ( ( n e. D |-> X ) e. T <-> ( ( n e. D |-> X ) : D --> A /\ ( n e. D |-> X ) finSupp (/) ) ) )
33	27, 30, 32	3bitr4d 300	1 |- ( ph -> ( ( n e. B |-> X ) e. S <-> ( n e. D |-> X ) e. T ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Oncon0 5723   Fun wfun 5882   -->wf 5884  (class class class)co 6650   supp csupp 7295   finSupp cfsupp 8275   CNF ccnf 8558

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-3an 1039  df-tru 1486  df-fal 1489  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-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-pred 5680  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-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-map 7859  df-fsupp 8276  df-cnf 8559

This theorem is referenced by:  cantnfres  8574