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Theorem recexprlemopu 6817

Description: The upper cut of 𝐵 is open. Lemma for recexpr 6828. (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 6813	. . 3 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
3		ltbtwnnqq 6605	. . . . . 6 ⊢ (𝑦 <_Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟))
4	3	biimpi 118	. . . . 5 ⊢ (𝑦 <_Q 𝑟 → ∃𝑞 ∈ Q (𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟))
5		simplr 496	. . . . . . . 8 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 <_Q 𝑟)
6		19.8a 1522	. . . . . . . . . 10 ⊢ ((𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)))
7	1	recexprlemelu 6813	. . . . . . . . . 10 ⊢ (𝑞 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑞 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
8	6, 7	sylibr 132	. . . . . . . . 9 ⊢ ((𝑦 <_Q 𝑞 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 ∈ (2^nd ‘𝐵))
9	8	adantlr 460	. . . . . . . 8 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 ∈ (2^nd ‘𝐵))
10	5, 9	jca 300	. . . . . . 7 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
11	10	expcom 114	. . . . . 6 ⊢ ((*_Q‘𝑦) ∈ (1^st ‘𝐴) → ((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) → (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵))))
12	11	reximdv 2462	. . . . 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 275	. . . 4 ⊢ ((𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
14	13	exlimiv 1529	. . 3 ⊢ (∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
15	2, 14	sylbi 119	. 2 ⊢ (𝑟 ∈ (2^nd ‘𝐵) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)))
16	15	3ad2ant3 961	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 102 ∧ w3a 919 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 ∃wrex 2349 ⟨cop 3401 class class class wbr 3785 ‘cfv 4922 1^st c1st 5785 2^nd c2nd 5786 Qcnq 6470 *_Qcrq 6474 <_Q cltq 6475 Pcnp 6481

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543

This theorem is referenced by: recexprlemrnd 6819

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