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Theorem recexprlemupu 7630

Description: The upper cut of 𝐵 is upper. Lemma for recexpr 7640. (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
recexprlemupu	⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)) → 𝑟 ∈ (2^nd ‘𝐵)))

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

**Proof of Theorem recexprlemupu**
Step	Hyp	Ref	Expression
1		ltsonq 7400	. . . . . . . . 9 ⊢ <_Q Or Q
2		ltrelnq 7367	. . . . . . . . 9 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	sotri 5026	. . . . . . . 8 ⊢ ((𝑦 <_Q 𝑞 ∧ 𝑞 <_Q 𝑟) → 𝑦 <_Q 𝑟)
4	3	expcom 116	. . . . . . 7 ⊢ (𝑞 <_Q 𝑟 → (𝑦 <_Q 𝑞 → 𝑦 <_Q 𝑟))
5	4	anim1d 336	. . . . . 6 ⊢ (𝑞 <_Q 𝑟 → ((𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → (𝑦 <_Q 𝑟 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
6	5	eximdv 1880	. . . . 5 ⊢ (𝑞 <_Q 𝑟 → (∃𝑦(𝑦 <_Q 𝑞 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑟 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
7		recexpr.1	. . . . . 6 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
8	7	recexprlemelu 7625	. . . . 5 ⊢ (𝑞 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑞 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
9	7	recexprlemelu 7625	. . . . 5 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
10	6, 8, 9	3imtr4g 205	. . . 4 ⊢ (𝑞 <_Q 𝑟 → (𝑞 ∈ (2^nd ‘𝐵) → 𝑟 ∈ (2^nd ‘𝐵)))
11	10	imp 124	. . 3 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)) → 𝑟 ∈ (2^nd ‘𝐵))
12	11	rexlimivw 2590	. 2 ⊢ (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)) → 𝑟 ∈ (2^nd ‘𝐵))
13	12	a1i 9	1 ⊢ ((𝐴 ∈ P ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐵)) → 𝑟 ∈ (2^nd ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∃wrex 2456 ⟨cop 3597 class class class wbr 4005 ‘cfv 5218 1^st c1st 6142 2^nd c2nd 6143 Qcnq 7282 *_Qcrq 7286 <_Q cltq 7287 Pcnp 7293

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-irdg 6374 df-oadd 6424 df-omul 6425 df-er 6538 df-ec 6540 df-qs 6544 df-ni 7306 df-mi 7308 df-lti 7309 df-enq 7349 df-nqqs 7350 df-ltnqqs 7355

This theorem is referenced by: recexprlemrnd 7631

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