Theorem opabfi 7108

Description: Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)

Hypotheses

Ref	Expression
opabfi.s	|- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }
opabfi.a	|- ( ph -> A e. Fin )
opabfi.b	|- ( ph -> B e. Fin )
opabfi.dc	|- ( ph -> A. x e. A A. y e. B DECID ps )

Assertion

Ref	Expression
opabfi	|- ( ph -> S e. Fin )

Distinct variable groups:   x, A, y   x, B, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   S( x, y)

Proof of Theorem opabfi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opabfi.a	. . 3 |- ( ph -> A e. Fin )
2		opabfi.b	. . 3 |- ( ph -> B e. Fin )
3		xpfi 7102	. . 3 |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin )
4	1, 2, 3	syl2anc 411	. 2 |- ( ph -> ( A X. B ) e. Fin )
5		opabfi.s	. . . 4 |- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }
6		opabssxp 4793	. . . 4 |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } C_ ( A X. B )
7	5, 6	eqsstri 3256	. . 3 |- S C_ ( A X. B )
8	7	a1i 9	. 2 |- ( ph -> S C_ ( A X. B ) )
9		xp2nd 6318	. . . . . 6 |- ( z e. ( A X. B ) -> ( 2nd ` z ) e. B )
10	9	adantl 277	. . . . 5 |- ( ( ph /\ z e. ( A X. B ) ) -> ( 2nd ` z ) e. B )
11		xp1st 6317	. . . . . . 7 |- ( z e. ( A X. B ) -> ( 1st ` z ) e. A )
12	11	adantl 277	. . . . . 6 |- ( ( ph /\ z e. ( A X. B ) ) -> ( 1st ` z ) e. A )
13		opabfi.dc	. . . . . . 7 |- ( ph -> A. x e. A A. y e. B DECID ps )
14	13	adantr 276	. . . . . 6 |- ( ( ph /\ z e. ( A X. B ) ) -> A. x e. A A. y e. B DECID ps )
15		nfcv 2372	. . . . . . . 8 |- F/_ x B
16		nfsbc1v 3047	. . . . . . . . 9 |- F/ x [. ( 1st ` z ) / x ]. ps
17	16	nfdc 1705	. . . . . . . 8 |- F/ xDECID [. ( 1st ` z ) / x ]. ps
18	15, 17	nfralw 2567	. . . . . . 7 |- F/ x A. y e. B DECID [. ( 1st ` z ) / x ]. ps
19		sbceq1a 3038	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( ps <-> [. ( 1st ` z ) / x ]. ps ) )
20	19	dcbid 843	. . . . . . . 8 |- ( x = ( 1st ` z ) -> (DECID ps <-> DECID [. ( 1st ` z ) / x ]. ps ) )
21	20	ralbidv 2530	. . . . . . 7 |- ( x = ( 1st ` z ) -> ( A. y e. B DECID ps <-> A. y e. B DECID [. ( 1st ` z ) / x ]. ps ) )
22	18, 21	rspc 2901	. . . . . 6 |- ( ( 1st ` z ) e. A -> ( A. x e. A A. y e. B DECID ps -> A. y e. B DECID [. ( 1st ` z ) / x ]. ps ) )
23	12, 14, 22	sylc 62	. . . . 5 |- ( ( ph /\ z e. ( A X. B ) ) -> A. y e. B DECID [. ( 1st ` z ) / x ]. ps )
24		nfsbc1v 3047	. . . . . . 7 |- F/ y [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
25	24	nfdc 1705	. . . . . 6 |- F/ yDECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
26		sbceq1a 3038	. . . . . . 7 |- ( y = ( 2nd ` z ) -> ( [. ( 1st ` z ) / x ]. ps <-> [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
27	26	dcbid 843	. . . . . 6 |- ( y = ( 2nd ` z ) -> (DECID [. ( 1st ` z ) / x ]. ps <-> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
28	25, 27	rspc 2901	. . . . 5 |- ( ( 2nd ` z ) e. B -> ( A. y e. B DECID [. ( 1st ` z ) / x ]. ps -> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
29	10, 23, 28	sylc 62	. . . 4 |- ( ( ph /\ z e. ( A X. B ) ) -> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps )
30		nfv 1574	. . . . . . . . . 10 |- F/ x ( ( 1st ` z ) e. A /\ y e. B )
31	30, 16	nfan 1611	. . . . . . . . 9 |- F/ x ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps )
32		nfv 1574	. . . . . . . . . 10 |- F/ y ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B )
33	32, 24	nfan 1611	. . . . . . . . 9 |- F/ y ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps )
34		eleq1 2292	. . . . . . . . . . 11 |- ( x = ( 1st ` z ) -> ( x e. A <-> ( 1st ` z ) e. A ) )
35	34	anbi1d 465	. . . . . . . . . 10 |- ( x = ( 1st ` z ) -> ( ( x e. A /\ y e. B ) <-> ( ( 1st ` z ) e. A /\ y e. B ) ) )
36	35, 19	anbi12d 473	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( ( ( x e. A /\ y e. B ) /\ ps ) <-> ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps ) ) )
37		eleq1 2292	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> ( y e. B <-> ( 2nd ` z ) e. B ) )
38	37	anbi2d 464	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) e. A /\ y e. B ) <-> ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) ) )
39	38, 26	anbi12d 473	. . . . . . . . 9 |- ( y = ( 2nd ` z ) -> ( ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps ) <-> ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) ) )
40	31, 33, 36, 39	opelopabgf 4358	. . . . . . . 8 |- ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } <-> ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) ) )
41	11, 9, 40	syl2anc 411	. . . . . . 7 |- ( z e. ( A X. B ) -> ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } <-> ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) ) )
42		1st2nd2 6327	. . . . . . . 8 |- ( z e. ( A X. B ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
43	5	a1i 9	. . . . . . . 8 |- ( z e. ( A X. B ) -> S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } )
44	42, 43	eleq12d 2300	. . . . . . 7 |- ( z e. ( A X. B ) -> ( z e. S <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } ) )
45		ibar 301	. . . . . . . 8 |- ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) -> ( [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps <-> ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) ) )
46	11, 9, 45	syl2anc 411	. . . . . . 7 |- ( z e. ( A X. B ) -> ( [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps <-> ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) ) )
47	41, 44, 46	3bitr4d 220	. . . . . 6 |- ( z e. ( A X. B ) -> ( z e. S <-> [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
48	47	dcbid 843	. . . . 5 |- ( z e. ( A X. B ) -> (DECID z e. S <-> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
49	48	adantl 277	. . . 4 |- ( ( ph /\ z e. ( A X. B ) ) -> (DECID z e. S <-> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
50	29, 49	mpbird 167	. . 3 |- ( ( ph /\ z e. ( A X. B ) ) -> DECID z e. S )
51	50	ralrimiva 2603	. 2 |- ( ph -> A. z e. ( A X. B )DECID z e. S )
52		ssfidc 7107	. 2 |- ( ( ( A X. B ) e. Fin /\ S C_ ( A X. B ) /\ A. z e. ( A X. B )DECID z e. S ) -> S e. Fin )
53	4, 8, 51, 52	syl3anc 1271	1 |- ( ph -> S e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   [.wsbc 3028   C_ wss 3197   <.cop 3669   {copab 4144   X. cxp 4717   ` cfv 5318   1stc1st 6290   2ndc2nd 6291   Fincfn 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6292  df-2nd 6293  df-1o 6568  df-er 6688  df-en 6896  df-fin 6898

This theorem is referenced by:  lgsquadlemsfi  15762  lgsquadlem3  15766