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Theorem recexprlemelu 7806

Description: Membership in the upper cut of 𝐵. Lemma for recexpr 7821. (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 2811	. 2 ⊢ (𝐶 ∈ (2^nd ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7548	. . . . . . 7 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4770	. . . . . 6 ⊢ (𝑦 <_Q 𝐶 → (𝑦 ∈ Q ∧ 𝐶 ∈ Q))
4	3	simprd 114	. . . . 5 ⊢ (𝑦 <_Q 𝐶 → 𝐶 ∈ Q)
5		elex 2811	. . . . 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 1644	. 2 ⊢ (∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝐶 ∈ V)
9		breq2 4086	. . . . 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 1871	. . 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 5629	. . . 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 7546	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4770	. . . . . . . . . 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 1644	. . . . . . 7 ⊢ (∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3299	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 4221	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))} ∈ V
21	2	brel 4770	. . . . . . . . . 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 1644	. . . . . . 7 ⊢ (∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3299	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 4221	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))} ∈ V
27	20, 26	op2nd 6291	. . . 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 2250	. . 3 ⊢ (2^nd ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))}
29	11, 28	elab2g 2950	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝐶 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))))
30	1, 8, 29	pm5.21nii 709	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 1395 ∃wex 1538 ∈ wcel 2200 {cab 2215 Vcvv 2799 ⟨cop 3669 class class class wbr 4082 ‘cfv 5317 1^st c1st 6282 2^nd c2nd 6283 Qcnq 7463 *_Qcrq 7467 <_Q cltq 7468

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-iinf 4679

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-2nd 6285 df-qs 6684 df-ni 7487 df-nqqs 7531 df-ltnqqs 7536

This theorem is referenced by: recexprlemm 7807 recexprlemopu 7810 recexprlemupu 7811 recexprlemdisj 7813 recexprlemloc 7814 recexprlem1ssu 7817 recexprlemss1u 7819

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