opabfi - Intuitionistic Logic Explorer

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Theorem opabfi 6992

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 6986	. . 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 4733	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ⊆ (𝐴 × 𝐵)
7	5, 6	eqsstri 3211	. . 3 ⊢ 𝑆 ⊆ (𝐴 × 𝐵)
8	7	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (𝐴 × 𝐵))
9		xp2nd 6219	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
10	9	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → (2^nd ‘𝑧) ∈ 𝐵)
11		xp1st 6218	. . . . . . 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 2336	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
16		nfsbc1v 3004	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1^st ‘𝑧) / 𝑥]𝜓
17	16	nfdc 1670	. . . . . . . 8 ⊢ Ⅎ𝑥DECID [(1^st ‘𝑧) / 𝑥]𝜓
18	15, 17	nfralw 2531	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓
19		sbceq1a 2995	. . . . . . . . 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 2494	. . . . . . 7 ⊢ (𝑥 = (1^st ‘𝑧) → (∀𝑦 ∈ 𝐵 DECID 𝜓 ↔ ∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓))
22	18, 21	rspc 2858	. . . . . 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 3004	. . . . . . 7 ⊢ Ⅎ𝑦[(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
25	24	nfdc 1670	. . . . . 6 ⊢ Ⅎ𝑦DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
26		sbceq1a 2995	. . . . . . 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 2858	. . . . 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 1539	. . . . . . . . . 10 ⊢ Ⅎ𝑥((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
31	30, 16	nfan 1576	. . . . . . . . 9 ⊢ Ⅎ𝑥(((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(1^st ‘𝑧) / 𝑥]𝜓)
32		nfv 1539	. . . . . . . . . 10 ⊢ Ⅎ𝑦((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
33	32, 24	nfan 1576	. . . . . . . . 9 ⊢ Ⅎ𝑦(((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)
34		eleq1 2256	. . . . . . . . . . 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 2256	. . . . . . . . . . 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 4300	. . . . . . . 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 6228	. . . . . . . 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 2264	. . . . . . 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 2567	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆)
52		ssfidc 6991	. 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 2164 ∀wral 2472 [wsbc 2985 ⊆ wss 3153 ⟨cop 3621 {copab 4089 × cxp 4657 ‘cfv 5254 1^st c1st 6191 2^nd c2nd 6192 Fincfn 6794

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-1st 6193 df-2nd 6194 df-1o 6469 df-er 6587 df-en 6795 df-fin 6797

This theorem is referenced by: lgsquadlem1 15191

W3C validator