Theorem opabfi 7042

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 7036	. . 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 4753	. . . 4 |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) } C_ ( A X. B )
7	5, 6	eqsstri 3226	. . 3 |- S C_ ( A X. B )
8	7	a1i 9	. 2 |- ( ph -> S C_ ( A X. B ) )
9		xp2nd 6259	. . . . . 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 6258	. . . . . . 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 2349	. . . . . . . 8 |- F/_ x B
16		nfsbc1v 3018	. . . . . . . . 9 |- F/ x [. ( 1st ` z ) / x ]. ps
17	16	nfdc 1683	. . . . . . . 8 |- F/ xDECID [. ( 1st ` z ) / x ]. ps
18	15, 17	nfralw 2544	. . . . . . 7 |- F/ x A. y e. B DECID [. ( 1st ` z ) / x ]. ps
19		sbceq1a 3009	. . . . . . . . 9 |- ( x = ( 1st ` z ) -> ( ps <-> [. ( 1st ` z ) / x ]. ps ) )
20	19	dcbid 840	. . . . . . . 8 |- ( x = ( 1st ` z ) -> (DECID ps <-> DECID [. ( 1st ` z ) / x ]. ps ) )
21	20	ralbidv 2507	. . . . . . 7 |- ( x = ( 1st ` z ) -> ( A. y e. B DECID ps <-> A. y e. B DECID [. ( 1st ` z ) / x ]. ps ) )
22	18, 21	rspc 2872	. . . . . 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 3018	. . . . . . 7 |- F/ y [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
25	24	nfdc 1683	. . . . . 6 |- F/ yDECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps
26		sbceq1a 3009	. . . . . . 7 |- ( y = ( 2nd ` z ) -> ( [. ( 1st ` z ) / x ]. ps <-> [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
27	26	dcbid 840	. . . . . 6 |- ( y = ( 2nd ` z ) -> (DECID [. ( 1st ` z ) / x ]. ps <-> DECID [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps ) )
28	25, 27	rspc 2872	. . . . 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 1552	. . . . . . . . . 10 |- F/ x ( ( 1st ` z ) e. A /\ y e. B )
31	30, 16	nfan 1589	. . . . . . . . 9 |- F/ x ( ( ( 1st ` z ) e. A /\ y e. B ) /\ [. ( 1st ` z ) / x ]. ps )
32		nfv 1552	. . . . . . . . . 10 |- F/ y ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B )
33	32, 24	nfan 1589	. . . . . . . . 9 |- F/ y ( ( ( 1st ` z ) e. A /\ ( 2nd ` z ) e. B ) /\ [. ( 2nd ` z ) / y ]. [. ( 1st ` z ) / x ]. ps )
34		eleq1 2269	. . . . . . . . . . 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 2269	. . . . . . . . . . 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 4320	. . . . . . . 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 6268	. . . . . . . 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 2277	. . . . . . 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 840	. . . . 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 2580	. 2 |- ( ph -> A. z e. ( A X. B )DECID z e. S )
52		ssfidc 7041	. 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 1250	1 |- ( ph -> S e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 836   = wceq 1373   e. wcel 2177   A.wral 2485   [.wsbc 2999   C_ wss 3167   <.cop 3637   {copab 4108   X. cxp 4677   ` cfv 5276   1stc1st 6231   2ndc2nd 6232   Fincfn 6834

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-1st 6233  df-2nd 6234  df-1o 6509  df-er 6627  df-en 6835  df-fin 6837

This theorem is referenced by:  lgsquadlemsfi  15596  lgsquadlem3  15600