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Theorem recexprlemopu 7770

Description: The upper cut of 𝐵 is open. Lemma for recexpr 7781. (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 7766	. . 3 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
3		ltbtwnnqq 7558	. . . . . 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 528	. . . . . . . 8 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 <_Q 𝑟)
6		19.8a 1614	. . . . . . . . . 10 ⊢ ((𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)))
7	1	recexprlemelu 7766	. . . . . . . . . 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 2608	. . . . 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 1622	. . 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 1023	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 981 = wceq 1373 ∃wex 1516 ∈ wcel 2177 {cab 2192 ∃wrex 2486 ⟨cop 3641 class class class wbr 4054 ‘cfv 5285 1^st c1st 6242 2^nd c2nd 6243 Qcnq 7423 *_Qcrq 7427 <_Q cltq 7428 Pcnp 7434

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4170 ax-sep 4173 ax-nul 4181 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-iinf 4649

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-ral 2490 df-rex 2491 df-reu 2492 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-int 3895 df-iun 3938 df-br 4055 df-opab 4117 df-mpt 4118 df-tr 4154 df-eprel 4349 df-id 4353 df-po 4356 df-iso 4357 df-iord 4426 df-on 4428 df-suc 4431 df-iom 4652 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-f1 5290 df-fo 5291 df-f1o 5292 df-fv 5293 df-ov 5965 df-oprab 5966 df-mpo 5967 df-1st 6244 df-2nd 6245 df-recs 6409 df-irdg 6474 df-1o 6520 df-oadd 6524 df-omul 6525 df-er 6638 df-ec 6640 df-qs 6644 df-ni 7447 df-pli 7448 df-mi 7449 df-lti 7450 df-plpq 7487 df-mpq 7488 df-enq 7490 df-nqqs 7491 df-plqqs 7492 df-mqqs 7493 df-1nqqs 7494 df-rq 7495 df-ltnqqs 7496

This theorem is referenced by: recexprlemrnd 7772

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