opabfi - Intuitionistic Logic Explorer

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Theorem opabfi 6999

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 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
opabfi.a	⊢ (𝜑 → 𝐴 ∈ Fin)
opabfi.b	⊢ (𝜑 → 𝐵 ∈ Fin)
opabfi.dc	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓)

**Assertion**
Ref	Expression
opabfi	⊢ (𝜑 → 𝑆 ∈ Fin)

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝜓(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 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		opabfi.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
3		xpfi 6993	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
4	1, 2, 3	syl2anc 411	. 2 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ Fin)
5		opabfi.s	. . . 4 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}
6		opabssxp 4737	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ⊆ (𝐴 × 𝐵)
7	5, 6	eqsstri 3215	. . 3 ⊢ 𝑆 ⊆ (𝐴 × 𝐵)
8	7	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (𝐴 × 𝐵))
9		xp2nd 6224	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
10	9	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → (2^nd ‘𝑧) ∈ 𝐵)
11		xp1st 6223	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
12	11	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → (1^st ‘𝑧) ∈ 𝐴)
13		opabfi.dc	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓)
14	13	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓)
15		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
16		nfsbc1v 3008	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1^st ‘𝑧) / 𝑥]𝜓
17	16	nfdc 1673	. . . . . . . 8 ⊢ Ⅎ𝑥DECID [(1^st ‘𝑧) / 𝑥]𝜓
18	15, 17	nfralw 2534	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓
19		sbceq1a 2999	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → (𝜓 ↔ [(1^st ‘𝑧) / 𝑥]𝜓))
20	19	dcbid 839	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → (DECID 𝜓 ↔ DECID [(1^st ‘𝑧) / 𝑥]𝜓))
21	20	ralbidv 2497	. . . . . . 7 ⊢ (𝑥 = (1^st ‘𝑧) → (∀𝑦 ∈ 𝐵 DECID 𝜓 ↔ ∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓))
22	18, 21	rspc 2862	. . . . . 6 ⊢ ((1^st ‘𝑧) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 DECID 𝜓 → ∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓))
23	12, 14, 22	sylc 62	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → ∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓)
24		nfsbc1v 3008	. . . . . . 7 ⊢ Ⅎ𝑦[(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
25	24	nfdc 1673	. . . . . 6 ⊢ Ⅎ𝑦DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
26		sbceq1a 2999	. . . . . . 7 ⊢ (𝑦 = (2^nd ‘𝑧) → ([(1^st ‘𝑧) / 𝑥]𝜓 ↔ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
27	26	dcbid 839	. . . . . 6 ⊢ (𝑦 = (2^nd ‘𝑧) → (DECID [(1^st ‘𝑧) / 𝑥]𝜓 ↔ DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
28	25, 27	rspc 2862	. . . . 5 ⊢ ((2^nd ‘𝑧) ∈ 𝐵 → (∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓 → DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
29	10, 23, 28	sylc 62	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)
30		nfv 1542	. . . . . . . . . 10 ⊢ Ⅎ𝑥((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
31	30, 16	nfan 1579	. . . . . . . . 9 ⊢ Ⅎ𝑥(((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(1^st ‘𝑧) / 𝑥]𝜓)
32		nfv 1542	. . . . . . . . . 10 ⊢ Ⅎ𝑦((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
33	32, 24	nfan 1579	. . . . . . . . 9 ⊢ Ⅎ𝑦(((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)
34		eleq1 2259	. . . . . . . . . . 11 ⊢ (𝑥 = (1^st ‘𝑧) → (𝑥 ∈ 𝐴 ↔ (1^st ‘𝑧) ∈ 𝐴))
35	34	anbi1d 465	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑧) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
36	35, 19	anbi12d 473	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ (((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(1^st ‘𝑧) / 𝑥]𝜓)))
37		eleq1 2259	. . . . . . . . . . 11 ⊢ (𝑦 = (2^nd ‘𝑧) → (𝑦 ∈ 𝐵 ↔ (2^nd ‘𝑧) ∈ 𝐵))
38	37	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑧) → (((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ ((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)))
39	38, 26	anbi12d 473	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → ((((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(1^st ‘𝑧) / 𝑥]𝜓) ↔ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)))
40	31, 33, 36, 39	opelopabgf 4304	. . . . . . . 8 ⊢ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → (⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ↔ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)))
41	11, 9, 40	syl2anc 411	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ↔ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)))
42		1st2nd2 6233	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
43	5	a1i 9	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)})
44	42, 43	eleq12d 2267	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝑧 ∈ 𝑆 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}))
45		ibar 301	. . . . . . . 8 ⊢ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ([(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓 ↔ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)))
46	11, 9, 45	syl2anc 411	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ([(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓 ↔ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)))
47	41, 44, 46	3bitr4d 220	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝑧 ∈ 𝑆 ↔ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
48	47	dcbid 839	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (DECID 𝑧 ∈ 𝑆 ↔ DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
49	48	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → (DECID 𝑧 ∈ 𝑆 ↔ DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
50	29, 49	mpbird 167	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → DECID 𝑧 ∈ 𝑆)
51	50	ralrimiva 2570	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆)
52		ssfidc 6998	. 2 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝑆 ⊆ (𝐴 × 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆) → 𝑆 ∈ Fin)
53	4, 8, 51, 52	syl3anc 1249	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2167 ∀wral 2475 [wsbc 2989 ⊆ wss 3157 ⟨cop 3625 {copab 4093 × cxp 4661 ‘cfv 5258 1^st c1st 6196 2^nd c2nd 6197 Fincfn 6799

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-1st 6198 df-2nd 6199 df-1o 6474 df-er 6592 df-en 6800 df-fin 6802

This theorem is referenced by: lgsquadlemsfi 15316 lgsquadlem3 15320

W3C validator