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Theorem recexprlemopu 7984

Description: The upper cut of 𝐵 is open. Lemma for recexpr 7995. (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 7980	. . 3 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
3		ltbtwnnqq 7772	. . . . . 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 533	. . . . . . . 8 ⊢ (((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)) → 𝑞 <_Q 𝑟)
6		19.8a 1643	. . . . . . . . . 10 ⊢ ((𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)))
7	1	recexprlemelu 7980	. . . . . . . . . 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 481	. . . . . . . 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 2651	. . . . 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 1651	. . 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 1051	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 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {cab 2224 ∃wrex 2529 ⟨cop 3708 class class class wbr 4125 ‘cfv 5372 1^st c1st 6362 2^nd c2nd 6363 Qcnq 7637 *_Qcrq 7641 <_Q cltq 7642 Pcnp 7648

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-eprel 4429 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-1o 6677 df-oadd 6681 df-omul 6682 df-er 6797 df-ec 6799 df-qs 6803 df-ni 7661 df-pli 7662 df-mi 7663 df-lti 7664 df-plpq 7701 df-mpq 7702 df-enq 7704 df-nqqs 7705 df-plqqs 7706 df-mqqs 7707 df-1nqqs 7708 df-rq 7709 df-ltnqqs 7710

This theorem is referenced by: recexprlemrnd 7986

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