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Theorem recexprlemlol 7999

Description: The lower cut of 𝐵 is lower. Lemma for recexpr 8011. (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
recexprlemlol	⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵)))

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

**Proof of Theorem recexprlemlol**
Step	Hyp	Ref	Expression
1		ltsonq 7771	. . . . . . . . 9 ⊢ <_Q Or Q
2		ltrelnq 7738	. . . . . . . . 9 ⊢ <_Q ⊆ (Q × Q)
3	1, 2	sotri 5186	. . . . . . . 8 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 <_Q 𝑦) → 𝑞 <_Q 𝑦)
4	3	ex 115	. . . . . . 7 ⊢ (𝑞 <_Q 𝑟 → (𝑟 <_Q 𝑦 → 𝑞 <_Q 𝑦))
5	4	anim1d 336	. . . . . 6 ⊢ (𝑞 <_Q 𝑟 → ((𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → (𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
6	5	eximdv 1933	. . . . 5 ⊢ (𝑞 <_Q 𝑟 → (∃𝑦(𝑟 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘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	recexprlemell 7995	. . . . 5 ⊢ (𝑟 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑟 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
9	7	recexprlemell 7995	. . . . 5 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
10	6, 8, 9	3imtr4g 205	. . . 4 ⊢ (𝑞 <_Q 𝑟 → (𝑟 ∈ (1^st ‘𝐵) → 𝑞 ∈ (1^st ‘𝐵)))
11	10	imp 124	. . 3 ⊢ ((𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵))
12	11	rexlimivw 2664	. 2 ⊢ (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵))
13	12	a1i 9	1 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐵)) → 𝑞 ∈ (1^st ‘𝐵)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {cab 2224 ∃wrex 2529 ⟨cop 3712 class class class wbr 4131 ‘cfv 5380 1^st c1st 6376 2^nd c2nd 6377 Qcnq 7653 *_Qcrq 7657 <_Q cltq 7658 Pcnp 7664

This proof depends on 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737

This proof 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 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-eprel 4436 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-oadd 6695 df-omul 6696 df-er 6811 df-ec 6813 df-qs 6817 df-ni 7677 df-mi 7679 df-lti 7680 df-enq 7720 df-nqqs 7721 df-ltnqqs 7726

This theorem is used by: recexprlemrnd 8002

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