Theorem opabfi 7137

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 7129	. . 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 4802	. . . 4 |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } C_ ( A X. B )
7	5, 6	eqsstri 3258	. . 3 |- S C_ ( A X. B )
8	7	a1i 9	. 2 |- ( ph -> S C_ ( A X. B ) )
9		xp2nd 6334	. . . . . 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 6333	. . . . . . 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 2373	. . . . . . . 8 |- F/_ x B
16		nfsbc1v 3049	. . . . . . . . 9 |- F/ x [. ( 1st ` z ) / x ]. ps
17	16	nfdc 1706	. . . . . . . 8 |- F/ xDECID [. ( 1st ` z ) / x ]. ps
18	15, 17	nfralw 2568	. . . . . . 7 |- F/ x A. y e. B DECID [. ( 1st ` z ) / x ]. ps
19		sbceq1a 3040	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( ps <-> [. ( 1st ` z ) / x ]. ps ) )
20	19	dcbid 845	. . . . . . . 8 |- ( x = ( 1st ` z ) -> (DECID ps <-> DECID [. ( 1st ` z ) / x ]. ps ) )
21	20	ralbidv 2531	. . . . . . 7 |- ( x = ( 1st ` z ) -> ( A. y e. B DECID ps <-> A. y e. B DECID [. ( 1st ` z ) / x ]. ps ) )
22	18, 21	rspc 2903	. . . . . 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 3049	. . . . . . 7 |- F/ y [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
25	24	nfdc 1706	. . . . . 6 |- F/ yDECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
26		sbceq1a 3040	. . . . . . 7 |- ( y = ( 2nd ` z ) -> ( [. ( 1st ` z ) / x ]. ps <-> [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
27	26	dcbid 845	. . . . . 6 |- ( y = ( 2nd ` z ) -> (DECID [. ( 1st ` z ) / x ]. ps <-> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
28	25, 27	rspc 2903	. . . . 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 1576	. . . . . . . . . 10 |- F/ x ( ( 1st ` z ) e. A /\ y e. B )
31	30, 16	nfan 1613	. . . . . . . . 9 |- F/ x ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps )
32		nfv 1576	. . . . . . . . . 10 |- F/ y ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B )
33	32, 24	nfan 1613	. . . . . . . . 9 |- F/ y ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps )
34		eleq1 2293	. . . . . . . . . . 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 2293	. . . . . . . . . . 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 4366	. . . . . . . 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 6343	. . . . . . . 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 2301	. . . . . . 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 845	. . . . 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 2604	. 2 |- ( ph -> A. z e. ( A X. B )DECID z e. S )
52		ssfidc 7135	. 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 1273	1 |- ( ph -> S e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 841   = wceq 1397   e. wcel 2201   A.wral 2509   [.wsbc 3030   C_ wss 3199   <.cop 3673   {copab 4150   X. cxp 4725   ` cfv 5328   1stc1st 6306   2ndc2nd 6307   Fincfn 6914

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-1st 6308  df-2nd 6309  df-1o 6587  df-er 6707  df-en 6915  df-fin 6917

This theorem is referenced by:  lgsquadlemsfi  15833  lgsquadlem3  15837