opabfi - Intuitionistic Logic Explorer

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Theorem opabfi 7068

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 7062	. . 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 4770	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ⊆ (𝐴 × 𝐵)
7	5, 6	eqsstri 3236	. . 3 ⊢ 𝑆 ⊆ (𝐴 × 𝐵)
8	7	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (𝐴 × 𝐵))
9		xp2nd 6282	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
10	9	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → (2^nd ‘𝑧) ∈ 𝐵)
11		xp1st 6281	. . . . . . 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 2352	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
16		nfsbc1v 3027	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1^st ‘𝑧) / 𝑥]𝜓
17	16	nfdc 1685	. . . . . . . 8 ⊢ Ⅎ𝑥DECID [(1^st ‘𝑧) / 𝑥]𝜓
18	15, 17	nfralw 2547	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓
19		sbceq1a 3018	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → (𝜓 ↔ [(1^st ‘𝑧) / 𝑥]𝜓))
20	19	dcbid 842	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → (DECID 𝜓 ↔ DECID [(1^st ‘𝑧) / 𝑥]𝜓))
21	20	ralbidv 2510	. . . . . . 7 ⊢ (𝑥 = (1^st ‘𝑧) → (∀𝑦 ∈ 𝐵 DECID 𝜓 ↔ ∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓))
22	18, 21	rspc 2881	. . . . . 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 3027	. . . . . . 7 ⊢ Ⅎ𝑦[(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
25	24	nfdc 1685	. . . . . 6 ⊢ Ⅎ𝑦DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
26		sbceq1a 3018	. . . . . . 7 ⊢ (𝑦 = (2^nd ‘𝑧) → ([(1^st ‘𝑧) / 𝑥]𝜓 ↔ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
27	26	dcbid 842	. . . . . 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 2881	. . . . 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 1554	. . . . . . . . . 10 ⊢ Ⅎ𝑥((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
31	30, 16	nfan 1591	. . . . . . . . 9 ⊢ Ⅎ𝑥(((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(1^st ‘𝑧) / 𝑥]𝜓)
32		nfv 1554	. . . . . . . . . 10 ⊢ Ⅎ𝑦((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
33	32, 24	nfan 1591	. . . . . . . . 9 ⊢ Ⅎ𝑦(((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)
34		eleq1 2272	. . . . . . . . . . 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 2272	. . . . . . . . . . 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 4337	. . . . . . . 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 6291	. . . . . . . 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 2280	. . . . . . 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 842	. . . . 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 2583	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆)
52		ssfidc 7067	. 2 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝑆 ⊆ (𝐴 × 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆) → 𝑆 ∈ Fin)
53	4, 8, 51, 52	syl3anc 1252	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 838 = wceq 1375 ∈ wcel 2180 ∀wral 2488 [wsbc 3008 ⊆ wss 3177 ⟨cop 3649 {copab 4123 × cxp 4694 ‘cfv 5294 1^st c1st 6254 2^nd c2nd 6255 Fincfn 6857

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-ral 2493 df-rex 2494 df-reu 2495 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-iord 4434 df-on 4436 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-1st 6256 df-2nd 6257 df-1o 6532 df-er 6650 df-en 6858 df-fin 6860

This theorem is referenced by: lgsquadlemsfi 15719 lgsquadlem3 15723

W3C validator