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Theorem recexprlemelu 7938

Description: Membership in the upper cut of 𝐵. Lemma for recexpr 7953. (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 2825	. 2 ⊢ (𝐶 ∈ (2^nd ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7680	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4802	. . . . . 6 ⊢ (𝑦 <_Q 𝐶 → (𝑦 ∈ Q ∧ 𝐶 ∈ Q))
4	3	simprd 114	. . . . 5 ⊢ (𝑦 <_Q 𝐶 → 𝐶 ∈ Q)
5		elex 2825	. . . . 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 1647	. 2 ⊢ (∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝐶 ∈ V)
9		breq2 4113	. . . . 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 1874	. . 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 5673	. . . 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 7678	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4802	. . . . . . . . . 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 1647	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3313	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4248	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ V
21	2	brel 4802	. . . . . . . . . 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 1647	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3313	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4248	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ V
27	20, 26	op2nd 6341	. . . 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 2253	. . 3 ⊢ (2^nd ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))}
29	11, 28	elab2g 2964	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))))
30	1, 8, 29	pm5.21nii 712	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 1398 ∃wex 1541 ∈ wcel 2203 {cab 2218 Vcvv 2813 ⟨cop 3692 class class class wbr 4109 ‘cfv 5352 1^st c1st 6332 2^nd c2nd 6333 Qcnq 7595 *_Qcrq 7599 <_Q cltq 7600

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-iinf 4710

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-2nd 6335 df-qs 6773 df-ni 7619 df-nqqs 7663 df-ltnqqs 7668

This theorem is referenced by: recexprlemm 7939 recexprlemopu 7942 recexprlemupu 7943 recexprlemdisj 7945 recexprlemloc 7946 recexprlem1ssu 7949 recexprlemss1u 7951

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