opabfi - Intuitionistic Logic Explorer

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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 ∈ 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 7036	. . 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 4753	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} ⊆ (𝐴 × 𝐵)
7	5, 6	eqsstri 3226	. . 3 ⊢ 𝑆 ⊆ (𝐴 × 𝐵)
8	7	a1i 9	. 2 ⊢ (𝜑 → 𝑆 ⊆ (𝐴 × 𝐵))
9		xp2nd 6259	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
10	9	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 × 𝐵)) → (2^nd ‘𝑧) ∈ 𝐵)
11		xp1st 6258	. . . . . . 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 2349	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
16		nfsbc1v 3018	. . . . . . . . 9 ⊢ Ⅎ𝑥[(1^st ‘𝑧) / 𝑥]𝜓
17	16	nfdc 1683	. . . . . . . 8 ⊢ Ⅎ𝑥DECID [(1^st ‘𝑧) / 𝑥]𝜓
18	15, 17	nfralw 2544	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓
19		sbceq1a 3009	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → (𝜓 ↔ [(1^st ‘𝑧) / 𝑥]𝜓))
20	19	dcbid 840	. . . . . . . 8 ⊢ (𝑥 = (1^st ‘𝑧) → (DECID 𝜓 ↔ DECID [(1^st ‘𝑧) / 𝑥]𝜓))
21	20	ralbidv 2507	. . . . . . 7 ⊢ (𝑥 = (1^st ‘𝑧) → (∀𝑦 ∈ 𝐵 DECID 𝜓 ↔ ∀𝑦 ∈ 𝐵 DECID [(1^st ‘𝑧) / 𝑥]𝜓))
22	18, 21	rspc 2872	. . . . . 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 3018	. . . . . . 7 ⊢ Ⅎ𝑦[(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
25	24	nfdc 1683	. . . . . 6 ⊢ Ⅎ𝑦DECID [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓
26		sbceq1a 3009	. . . . . . 7 ⊢ (𝑦 = (2^nd ‘𝑧) → ([(1^st ‘𝑧) / 𝑥]𝜓 ↔ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓))
27	26	dcbid 840	. . . . . 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 2872	. . . . 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 1552	. . . . . . . . . 10 ⊢ Ⅎ𝑥((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
31	30, 16	nfan 1589	. . . . . . . . 9 ⊢ Ⅎ𝑥(((1^st ‘𝑧) ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ [(1^st ‘𝑧) / 𝑥]𝜓)
32		nfv 1552	. . . . . . . . . 10 ⊢ Ⅎ𝑦((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵)
33	32, 24	nfan 1589	. . . . . . . . 9 ⊢ Ⅎ𝑦(((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) ∧ [(2^nd ‘𝑧) / 𝑦][(1^st ‘𝑧) / 𝑥]𝜓)
34		eleq1 2269	. . . . . . . . . . 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 2269	. . . . . . . . . . 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 4320	. . . . . . . 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 6268	. . . . . . . 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 2277	. . . . . . 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 840	. . . . 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 2580	. 2 ⊢ (𝜑 → ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆)
52		ssfidc 7041	. 2 ⊢ (((𝐴 × 𝐵) ∈ Fin ∧ 𝑆 ⊆ (𝐴 × 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)DECID 𝑧 ∈ 𝑆) → 𝑆 ∈ Fin)
53	4, 8, 51, 52	syl3anc 1250	1 ⊢ (𝜑 → 𝑆 ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 836 = wceq 1373 ∈ wcel 2177 ∀wral 2485 [wsbc 2999 ⊆ wss 3167 ⟨cop 3637 {copab 4108 × cxp 4677 ‘cfv 5276 1^st c1st 6231 2^nd c2nd 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

W3C validator