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Theorem recexprlemelu 7657

Description: Membership in the upper cut of 𝐵. Lemma for recexpr 7672. (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
recexprlemelu	⊢ (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))

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

**Proof of Theorem recexprlemelu**
Step	Hyp	Ref	Expression
1		elex 2763	. 2 ⊢ (𝐶 ∈ (2^nd ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7399	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4699	. . . . . 6 ⊢ (𝑦 <_Q 𝐶 → (𝑦 ∈ Q ∧ 𝐶 ∈ Q))
4	3	simprd 114	. . . . 5 ⊢ (𝑦 <_Q 𝐶 → 𝐶 ∈ Q)
5		elex 2763	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 ∈ V)
6	4, 5	syl 14	. . . 4 ⊢ (𝑦 <_Q 𝐶 → 𝐶 ∈ V)
7	6	adantr 276	. . 3 ⊢ ((𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1609	. 2 ⊢ (∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝐶 ∈ V)
9		breq2 4025	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑦 <_Q 𝑥 ↔ 𝑦 <_Q 𝐶))
10	9	anbi1d 465	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) ↔ (𝑦 <_Q 𝐶 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
11	10	exbidv 1836	. . 3 ⊢ (𝑥 = 𝐶 → (∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (_Q‘𝑦) ∈ (1^st ‘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 5540	. . . 4 ⊢ (2^nd ‘𝐵) = (2^nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩)
14		nqex 7397	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4699	. . . . . . . . . 10 ⊢ (𝑥 <_Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
16	15	simpld 112	. . . . . . . . 9 ⊢ (𝑥 <_Q 𝑦 → 𝑥 ∈ Q)
17	16	adantr 276	. . . . . . . 8 ⊢ ((𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1609	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3245	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4159	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ V
21	2	brel 4699	. . . . . . . . . 10 ⊢ (𝑦 <_Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
22	21	simprd 114	. . . . . . . . 9 ⊢ (𝑦 <_Q 𝑥 → 𝑥 ∈ Q)
23	22	adantr 276	. . . . . . . 8 ⊢ ((𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1609	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3245	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4159	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ V
27	20, 26	op2nd 6176	. . . 4 ⊢ (2^nd ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))}
28	13, 27	eqtri 2210	. . 3 ⊢ (2^nd ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))}
29	11, 28	elab2g 2899	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))))
30	1, 8, 29	pm5.21nii 705	1 ⊢ (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 {cab 2175 Vcvv 2752 ⟨cop 3613 class class class wbr 4021 ‘cfv 5238 1^st c1st 6167 2^nd c2nd 6168 Qcnq 7314 *_Qcrq 7318 <_Q cltq 7319

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 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-iinf 4608

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-2nd 6170 df-qs 6569 df-ni 7338 df-nqqs 7382 df-ltnqqs 7387

This theorem is referenced by: recexprlemm 7658 recexprlemopu 7661 recexprlemupu 7662 recexprlemdisj 7664 recexprlemloc 7665 recexprlem1ssu 7668 recexprlemss1u 7670

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