recexprlemell - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > recexprlemell		GIF version

Theorem recexprlemell 6812

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 6828. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlemell	⊢ (𝐶 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦

**Proof of Theorem recexprlemell**
Step	Hyp	Ref	Expression
1		elex 2610	. 2 ⊢ (𝐶 ∈ (1^st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 6555	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4410	. . . . . 6 ⊢ (𝐶 <_Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 110	. . . . 5 ⊢ (𝐶 <_Q 𝑦 → 𝐶 ∈ Q)
5		elex 2610	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 ∈ V)
6	4, 5	syl 14	. . . 4 ⊢ (𝐶 <_Q 𝑦 → 𝐶 ∈ V)
7	6	adantr 270	. . 3 ⊢ ((𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1529	. 2 ⊢ (∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 3788	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 <_Q 𝑦 ↔ 𝐶 <_Q 𝑦))
10	9	anbi1d 452	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) ↔ (𝐶 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
11	10	exbidv 1746	. . 3 ⊢ (𝑥 = 𝐶 → (∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
12		recexpr.1	. . . . 5 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
13	12	fveq2i 5201	. . . 4 ⊢ (1^st ‘𝐵) = (1^st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩)
14		nqex 6553	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4410	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
16	15	simpld 110	. . . . . . . . 9 ⊢ (𝑥 <_Q 𝑦 → 𝑥 ∈ Q)
17	16	adantr 270	. . . . . . . 8 ⊢ ((𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1529	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3069	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 3916	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ V
21	2	brel 4410	. . . . . . . . . 10 ⊢ (𝑦 <_Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
22	21	simprd 112	. . . . . . . . 9 ⊢ (𝑦 <_Q 𝑥 → 𝑥 ∈ Q)
23	22	adantr 270	. . . . . . . 8 ⊢ ((𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1529	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3069	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 3916	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ V
27	20, 26	op1st 5793	. . . 4 ⊢ (1^st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))}
28	13, 27	eqtri 2101	. . 3 ⊢ (1^st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))}
29	11, 28	elab2g 2740	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 652	1 ⊢ (𝐶 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝐶 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))

Colors of variables: wff set class

Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 Vcvv 2601 ⟨cop 3401 class class class wbr 3785 ‘cfv 4922 1^st c1st 5785 2^nd c2nd 5786 Qcnq 6470 *_Qcrq 6474 <_Q cltq 6475

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1st 5787 df-qs 6135 df-ni 6494 df-nqqs 6538 df-ltnqqs 6543

This theorem is referenced by: recexprlemm 6814 recexprlemopl 6815 recexprlemlol 6816 recexprlemdisj 6820 recexprlemloc 6821 recexprlem1ssl 6823 recexprlemss1l 6825

W3C validator