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Theorem recexprlemopu 7942

Description: The upper cut of 𝐵 is open. Lemma for recexpr 7953. (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 7938	. . 3 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
3		ltbtwnnqq 7730	. . . . . 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 1639	. . . . . . . . . 10 ⊢ ((𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)))
7	1	recexprlemelu 7938	. . . . . . . . . 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 2643	. . . . 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 1647	. . 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 1047	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 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2203 {cab 2218 ∃wrex 2521 ⟨cop 3692 class class class wbr 4109 ‘cfv 5352 1^st c1st 6332 2^nd c2nd 6333 Qcnq 7595 *_Qcrq 7599 <_Q cltq 7600 Pcnp 7606

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-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 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-ne 2413 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-nul 3509 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-tr 4209 df-eprel 4410 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-suc 4492 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-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-1o 6647 df-oadd 6651 df-omul 6652 df-er 6767 df-ec 6769 df-qs 6773 df-ni 7619 df-pli 7620 df-mi 7621 df-lti 7622 df-plpq 7659 df-mpq 7660 df-enq 7662 df-nqqs 7663 df-plqqs 7664 df-mqqs 7665 df-1nqqs 7666 df-rq 7667 df-ltnqqs 7668

This theorem is referenced by: recexprlemrnd 7944

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