Theorem mapfien2 8314

Description: Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)

Hypotheses

Ref	Expression
mapfien2.s	|- S = { x e. ( B ^m A ) | x finSupp .0. }
mapfien2.t	|- T = { x e. ( D ^m C ) | x finSupp W }
mapfien2.ac	|- ( ph -> A ~~ C )
mapfien2.bd	|- ( ph -> B ~~ D )
mapfien2.z	|- ( ph -> .0. e. B )
mapfien2.w	|- ( ph -> W e. D )

Assertion

Ref	Expression
mapfien2	|- ( ph -> S ~~ T )

Distinct variable groups:   x, A   x, B   x, C   x, D   x, .0.   x, W

Allowed substitution hints:   ph( x)   S( x)   T( x)

Proof of Theorem mapfien2

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapfien2.z	. . 3 |- ( ph -> .0. e. B )
2		mapfien2.w	. . 3 |- ( ph -> W e. D )
3		mapfien2.bd	. . 3 |- ( ph -> B ~~ D )
4		enfixsn 8069	. . 3 |- ( ( .0. e. B /\ W e. D /\ B ~~ D ) -> E. y ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) )
5	1, 2, 3, 4	syl3anc 1326	. 2 |- ( ph -> E. y ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) )
6		mapfien2.ac	. . . . 5 |- ( ph -> A ~~ C )
7		bren 7964	. . . . 5 |- ( A ~~ C <-> E. z z : A -1-1-onto-> C )
8	6, 7	sylib 208	. . . 4 |- ( ph -> E. z z : A -1-1-onto-> C )
9		mapfien2.s	. . . . . . . . . 10 |- S = { x e. ( B ^m A ) | x finSupp .0. }
10		eqid 2622	. . . . . . . . . 10 |- { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } = { x e. ( D ^m C ) | x finSupp ( y ` .0. ) }
11		eqid 2622	. . . . . . . . . 10 |- ( y ` .0. ) = ( y ` .0. )
12		f1ocnv 6149	. . . . . . . . . . 11 |- ( z : A -1-1-onto-> C -> `' z : C -1-1-onto-> A )
13	12	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> `' z : C -1-1-onto-> A )
14		simp3 1063	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> y : B -1-1-onto-> D )
15	6	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> A ~~ C )
16		relen 7960	. . . . . . . . . . . 12 |- Rel ~~
17	16	brrelexi 5158	. . . . . . . . . . 11 |- ( A ~~ C -> A e. _V )
18	15, 17	syl 17	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> A e. _V )
19	3	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> B ~~ D )
20	16	brrelexi 5158	. . . . . . . . . . 11 |- ( B ~~ D -> B e. _V )
21	19, 20	syl 17	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> B e. _V )
22	16	brrelex2i 5159	. . . . . . . . . . 11 |- ( A ~~ C -> C e. _V )
23	15, 22	syl 17	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> C e. _V )
24	16	brrelex2i 5159	. . . . . . . . . . 11 |- ( B ~~ D -> D e. _V )
25	19, 24	syl 17	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> D e. _V )
26	1	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> .0. e. B )
27	9, 10, 11, 13, 14, 18, 21, 23, 25, 26	mapfien 8313	. . . . . . . . 9 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> ( w e. S |-> ( y o. ( w o. `' z ) ) ) : S -1-1-onto-> { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } )
28		ovex 6678	. . . . . . . . . . 11 |- ( B ^m A ) e. _V
29	9, 28	rabex2 4815	. . . . . . . . . 10 |- S e. _V
30	29	f1oen 7976	. . . . . . . . 9 |- ( ( w e. S |-> ( y o. ( w o. `' z ) ) ) : S -1-1-onto-> { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } -> S ~~ { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } )
31	27, 30	syl 17	. . . . . . . 8 |- ( ( ph /\ z : A -1-1-onto-> C /\ y : B -1-1-onto-> D ) -> S ~~ { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } )
32	31	3adant3r 1323	. . . . . . 7 |- ( ( ph /\ z : A -1-1-onto-> C /\ ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) ) -> S ~~ { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } )
33		breq2 4657	. . . . . . . . . . 11 |- ( ( y ` .0. ) = W -> ( x finSupp ( y ` .0. ) <-> x finSupp W ) )
34	33	rabbidv 3189	. . . . . . . . . 10 |- ( ( y ` .0. ) = W -> { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } = { x e. ( D ^m C ) | x finSupp W } )
35		mapfien2.t	. . . . . . . . . 10 |- T = { x e. ( D ^m C ) | x finSupp W }
36	34, 35	syl6eqr 2674	. . . . . . . . 9 |- ( ( y ` .0. ) = W -> { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } = T )
37	36	adantl 482	. . . . . . . 8 |- ( ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) -> { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } = T )
38	37	3ad2ant3 1084	. . . . . . 7 |- ( ( ph /\ z : A -1-1-onto-> C /\ ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) ) -> { x e. ( D ^m C ) | x finSupp ( y ` .0. ) } = T )
39	32, 38	breqtrd 4679	. . . . . 6 |- ( ( ph /\ z : A -1-1-onto-> C /\ ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) ) -> S ~~ T )
40	39	3exp 1264	. . . . 5 |- ( ph -> ( z : A -1-1-onto-> C -> ( ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) -> S ~~ T ) ) )
41	40	exlimdv 1861	. . . 4 |- ( ph -> ( E. z z : A -1-1-onto-> C -> ( ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) -> S ~~ T ) ) )
42	8, 41	mpd 15	. . 3 |- ( ph -> ( ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) -> S ~~ T ) )
43	42	exlimdv 1861	. 2 |- ( ph -> ( E. y ( y : B -1-1-onto-> D /\ ( y ` .0. ) = W ) -> S ~~ T ) )
44	5, 43	mpd 15	1 |- ( ph -> S ~~ T )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   o. ccom 5118   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   ~~ cen 7952   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-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-iun 4522  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-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-supp 7296  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-fsupp 8276

This theorem is referenced by:  frlmpwfi  37668