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Theorem recexprlemopu 7846

Description: The upper cut of 𝐵 is open. Lemma for recexpr 7857. (Contributed by Jim Kingdon, 28-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
recexprlemopu	⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2^nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))

Distinct variable groups: 𝑟,𝑞,𝑥,𝑦,𝐴 𝐵,𝑞,𝑟,𝑥,𝑦

**Proof of Theorem recexprlemopu**
Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemelu 7842	. . 3 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
3		ltbtwnnqq 7634	. . . . . 6 ⊢ (𝑦 <_Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟))
4	3	biimpi 120	. . . . 5 ⊢ (𝑦 <_Q 𝑟 → ∃𝑞 ∈ Q (𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟))
5		simplr 529	. . . . . . . 8 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 <_Q 𝑟)
6		19.8a 1638	. . . . . . . . . 10 ⊢ ((𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)))
7	1	recexprlemelu 7842	. . . . . . . . . 10 ⊢ (𝑞 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑞 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
8	6, 7	sylibr 134	. . . . . . . . 9 ⊢ ((𝑦 <_Q 𝑞 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 ∈ (2^nd ‘𝐵))
9	8	adantlr 477	. . . . . . . 8 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 ∈ (2^nd ‘𝐵))
10	5, 9	jca 306	. . . . . . 7 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
11	10	expcom 116	. . . . . 6 ⊢ ((*_Q‘𝑦) ∈ (1^st ‘𝐴) → ((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) → (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵))))
12	11	reximdv 2633	. . . . 5 ⊢ ((*_Q‘𝑦) ∈ (1^st ‘𝐴) → (∃𝑞 ∈ Q (𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵))))
13	4, 12	mpan9 281	. . . 4 ⊢ ((𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
14	13	exlimiv 1646	. . 3 ⊢ (∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
15	2, 14	sylbi 121	. 2 ⊢ (𝑟 ∈ (2^nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
16	15	3ad2ant3 1046	1 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2^nd ‘𝐵)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1004 = wceq 1397 ∃wex 1540 ∈ wcel 2202 {cab 2217 ∃wrex 2511 ⟨cop 3672 class class class wbr 4088 ‘cfv 5326 1^st c1st 6300 2^nd c2nd 6301 Qcnq 7499 *_Qcrq 7503 <_Q cltq 7504 Pcnp 7510

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-eprel 4386 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-1o 6581 df-oadd 6585 df-omul 6586 df-er 6701 df-ec 6703 df-qs 6707 df-ni 7523 df-pli 7524 df-mi 7525 df-lti 7526 df-plpq 7563 df-mpq 7564 df-enq 7566 df-nqqs 7567 df-plqqs 7568 df-mqqs 7569 df-1nqqs 7570 df-rq 7571 df-ltnqqs 7572

This theorem is referenced by: recexprlemrnd 7848

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