Theorem opabfi 7241

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 7233	. . 3 |- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin )
4	1, 2, 3	syl2anc 415	. 2 |- ( ph -> ( A X. B ) e. Fin )
5		opabfi.s	. . . 4 |- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }
6		opabssxp 4847	. . . 4 |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } C_ ( A X. B )
7	5, 6	eqsstri 3280	. . 3 |- S C_ ( A X. B )
8	7	a1i 9	. 2 |- ( ph -> S C_ ( A X. B ) )
9		xp2nd 6394	. . . . . 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 6393	. . . . . . 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 2392	. . . . . . . 8 |- F/_ x B
16		nfsbc1v 3070	. . . . . . . . 9 |- F/ x [. ( 1st ` z ) / x ]. ps
17	16	nfdc 1711	. . . . . . . 8 |- F/ xDECID [. ( 1st ` z ) / x ]. ps
18	15, 17	nfralw 2587	. . . . . . 7 |- F/ x A. y e. B DECID [. ( 1st ` z ) / x ]. ps
19		sbceq1a 3061	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( ps <-> [. ( 1st ` z ) / x ]. ps ) )
20	19	dcbid 850	. . . . . . . 8 |- ( x = ( 1st ` z ) -> (DECID ps <-> DECID [. ( 1st ` z ) / x ]. ps ) )
21	20	ralbidv 2550	. . . . . . 7 |- ( x = ( 1st ` z ) -> ( A. y e. B DECID ps <-> A. y e. B DECID [. ( 1st ` z ) / x ]. ps ) )
22	18, 21	rspc 2923	. . . . . 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 3070	. . . . . . 7 |- F/ y [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
25	24	nfdc 1711	. . . . . 6 |- F/ yDECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
26		sbceq1a 3061	. . . . . . 7 |- ( y = ( 2nd ` z ) -> ( [. ( 1st ` z ) / x ]. ps <-> [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
27	26	dcbid 850	. . . . . 6 |- ( y = ( 2nd ` z ) -> (DECID [. ( 1st ` z ) / x ]. ps <-> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
28	25, 27	rspc 2923	. . . . 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 1581	. . . . . . . . . 10 |- F/ x ( ( 1st ` z ) e. A /\ y e. B )
31	30, 16	nfan 1618	. . . . . . . . 9 |- F/ x ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps )
32		nfv 1581	. . . . . . . . . 10 |- F/ y ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B )
33	32, 24	nfan 1618	. . . . . . . . 9 |- F/ y ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps )
34		eleq1 2301	. . . . . . . . . . 11 |- ( x = ( 1st ` z ) -> ( x e. A <-> ( 1st ` z ) e. A ) )
35	34	anbi1d 469	. . . . . . . . . 10 |- ( x = ( 1st ` z ) -> ( ( x e. A /\ y e. B ) <-> ( ( 1st ` z ) e. A /\ y e. B ) ) )
36	35, 19	anbi12d 477	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( ( ( x e. A /\ y e. B ) /\ ps ) <-> ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps ) ) )
37		eleq1 2301	. . . . . . . . . . 11 |- ( y = ( 2nd ` z ) -> ( y e. B <-> ( 2nd ` z ) e. B ) )
38	37	anbi2d 468	. . . . . . . . . 10 |- ( y = ( 2nd ` z ) -> ( ( ( 1st ` z ) e. A /\ y e. B ) <-> ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) ) )
39	38, 26	anbi12d 477	. . . . . . . . 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 4410	. . . . . . . 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 415	. . . . . . 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 6403	. . . . . . . 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 2309	. . . . . . 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 415	. . . . . . 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 850	. . . . 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 2623	. 2 |- ( ph -> A. z e. ( A X. B )DECID z e. S )
52		ssfidc 7239	. 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 1278	1 |- ( ph -> S e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   [.wsbc 3051   C_ wss 3220   <.cop 3711   {copab 4189   X. cxp 4770   ` cfv 5375   1stc1st 6366   2ndc2nd 6367   Fincfn 7016

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6368  df-2nd 6369  df-1o 6681  df-er 6801  df-en 7017  df-fin 7019

This theorem is referenced by:  lgsquadlemsfi  16177  lgsquadlem3  16181