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Theorem recexprlemupu 7776

Description: The upper cut of 𝐵 is upper. Lemma for recexpr 7786. (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 7546	. . . . . . . . 9 ⊢ <_Q Or Q
2		ltrelnq 7513	. . . . . . . . 9 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	sotri 5097	. . . . . . . 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 1904	. . . . 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 7771	. . . . 5 ⊢ (𝑞 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑞 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
9	7	recexprlemelu 7771	. . . . 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 2621	. 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 1373 ∃wex 1516 ∈ wcel 2178 {cab 2193 ∃wrex 2487 ⟨cop 3646 class class class wbr 4059 ‘cfv 5290 1^st c1st 6247 2^nd c2nd 6248 Qcnq 7428 *_Qcrq 7432 <_Q cltq 7433 Pcnp 7439

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 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654

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 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-eprel 4354 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-oadd 6529 df-omul 6530 df-er 6643 df-ec 6645 df-qs 6649 df-ni 7452 df-mi 7454 df-lti 7455 df-enq 7495 df-nqqs 7496 df-ltnqqs 7501

This theorem is referenced by: recexprlemrnd 7777

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