Theorem suppeqfsuppbi 8289

Description: If two functions have the same support, one function is finitely supported iff the other one is finitely supported. (Contributed by AV, 30-Jun-2019.)

Assertion

Ref	Expression
suppeqfsuppbi	|- ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) -> ( ( F supp Z ) = ( G supp Z ) -> ( F finSupp Z <-> G finSupp Z ) ) )

Proof of Theorem suppeqfsuppbi

Step	Hyp	Ref	Expression
1		simprlr 803	. . . . . 6 |- ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) -> Fun F )
2		simprll 802	. . . . . 6 |- ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) -> F e. U )
3		simpl 473	. . . . . 6 |- ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) -> Z e. _V )
4		funisfsupp 8280	. . . . . 6 |- ( ( Fun F /\ F e. U /\ Z e. _V ) -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
5	1, 2, 3, 4	syl3anc 1326	. . . . 5 |- ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
6	5	adantr 481	. . . 4 |- ( ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) /\ ( F supp Z ) = ( G supp Z ) ) -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
7		simpr 477	. . . . . . . . . 10 |- ( ( G e. V /\ Fun G ) -> Fun G )
8	7	adantr 481	. . . . . . . . 9 |- ( ( ( G e. V /\ Fun G ) /\ Z e. _V ) -> Fun G )
9		simpl 473	. . . . . . . . . 10 |- ( ( G e. V /\ Fun G ) -> G e. V )
10	9	adantr 481	. . . . . . . . 9 |- ( ( ( G e. V /\ Fun G ) /\ Z e. _V ) -> G e. V )
11		simpr 477	. . . . . . . . 9 |- ( ( ( G e. V /\ Fun G ) /\ Z e. _V ) -> Z e. _V )
12		funisfsupp 8280	. . . . . . . . 9 |- ( ( Fun G /\ G e. V /\ Z e. _V ) -> ( G finSupp Z <-> ( G supp Z ) e. Fin ) )
13	8, 10, 11, 12	syl3anc 1326	. . . . . . . 8 |- ( ( ( G e. V /\ Fun G ) /\ Z e. _V ) -> ( G finSupp Z <-> ( G supp Z ) e. Fin ) )
14	13	ex 450	. . . . . . 7 |- ( ( G e. V /\ Fun G ) -> ( Z e. _V -> ( G finSupp Z <-> ( G supp Z ) e. Fin ) ) )
15	14	adantl 482	. . . . . 6 |- ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) -> ( Z e. _V -> ( G finSupp Z <-> ( G supp Z ) e. Fin ) ) )
16	15	impcom 446	. . . . 5 |- ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) -> ( G finSupp Z <-> ( G supp Z ) e. Fin ) )
17		eleq1 2689	. . . . . 6 |- ( ( F supp Z ) = ( G supp Z ) -> ( ( F supp Z ) e. Fin <-> ( G supp Z ) e. Fin ) )
18	17	bicomd 213	. . . . 5 |- ( ( F supp Z ) = ( G supp Z ) -> ( ( G supp Z ) e. Fin <-> ( F supp Z ) e. Fin ) )
19	16, 18	sylan9bb 736	. . . 4 |- ( ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) /\ ( F supp Z ) = ( G supp Z ) ) -> ( G finSupp Z <-> ( F supp Z ) e. Fin ) )
20	6, 19	bitr4d 271	. . 3 |- ( ( ( Z e. _V /\ ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) ) /\ ( F supp Z ) = ( G supp Z ) ) -> ( F finSupp Z <-> G finSupp Z ) )
21	20	exp31 630	. 2 |- ( Z e. _V -> ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) -> ( ( F supp Z ) = ( G supp Z ) -> ( F finSupp Z <-> G finSupp Z ) ) ) )
22		relfsupp 8277	. . . . 5 |- Rel finSupp
23	22	brrelex2i 5159	. . . 4 |- ( F finSupp Z -> Z e. _V )
24	22	brrelex2i 5159	. . . 4 |- ( G finSupp Z -> Z e. _V )
25	23, 24	pm5.21ni 367	. . 3 |- ( -. Z e. _V -> ( F finSupp Z <-> G finSupp Z ) )
26	25	2a1d 26	. 2 |- ( -. Z e. _V -> ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) -> ( ( F supp Z ) = ( G supp Z ) -> ( F finSupp Z <-> G finSupp Z ) ) ) )
27	21, 26	pm2.61i 176	1 |- ( ( ( F e. U /\ Fun F ) /\ ( G e. V /\ Fun G ) ) -> ( ( F supp Z ) = ( G supp Z ) -> ( F finSupp Z <-> G finSupp Z ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   Fun wfun 5882  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-fsupp 8276

This theorem is referenced by:  cantnfrescl  8573