Theorem recexprlemell 7244

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7260. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩

Assertion

Ref	Expression
recexprlemell	⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem recexprlemell

Step	Hyp	Ref	Expression
1		elex 2633	. 2 ⊢ (𝐶 ∈ (1st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 6987	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4505	. . . . . 6 ⊢ (𝐶 <Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 111	. . . . 5 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ Q)
5		elex 2633	. . . . 5 ⊢ (𝐶 ∈ Q → 𝐶 ∈ V)
6	4, 5	syl 14	. . . 4 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ V)
7	6	adantr 271	. . 3 ⊢ ((𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
8	7	exlimiv 1535	. 2 ⊢ (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 3856	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 <Q 𝑦 ↔ 𝐶 <Q 𝑦))
10	9	anbi1d 454	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ (𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
11	10	exbidv 1754	. . 3 ⊢ (𝑥 = 𝐶 → (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
12		recexpr.1	. . . . 5 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
13	12	fveq2i 5323	. . . 4 ⊢ (1st ‘𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩)
14		nqex 6985	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4505	. . . . . . . . . 10 ⊢ (𝑥 <Q 𝑦 → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
16	15	simpld 111	. . . . . . . . 9 ⊢ (𝑥 <Q 𝑦 → 𝑥 ∈ Q)
17	16	adantr 271	. . . . . . . 8 ⊢ ((𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
18	17	exlimiv 1535	. . . . . . 7 ⊢ (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3099	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 3985	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ∈ V
21	2	brel 4505	. . . . . . . . . 10 ⊢ (𝑦 <Q 𝑥 → (𝑦 ∈ Q ∧ 𝑥 ∈ Q))
22	21	simprd 113	. . . . . . . . 9 ⊢ (𝑦 <Q 𝑥 → 𝑥 ∈ Q)
23	22	adantr 271	. . . . . . . 8 ⊢ ((𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
24	23	exlimiv 1535	. . . . . . 7 ⊢ (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3099	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 3985	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ∈ V
27	20, 26	op1st 5933	. . . 4 ⊢ (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
28	13, 27	eqtri 2109	. . 3 ⊢ (1st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
29	11, 28	elab2g 2765	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 656	1 ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1290  ∃wex 1427   ∈ wcel 1439  {cab 2075  Vcvv 2622  ⟨cop 3455   class class class wbr 3853  ‘cfv 5030  1^st c1st 5925  2^nd c2nd 5926  Qcnq 6902  *_Qcrq 6906   <_Q cltq 6907

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-iinf 4418

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-1st 5927  df-qs 6314  df-ni 6926  df-nqqs 6970  df-ltnqqs 6975

This theorem is referenced by:  recexprlemm  7246  recexprlemopl  7247  recexprlemlol  7248  recexprlemdisj  7252  recexprlemloc  7253  recexprlem1ssl  7255  recexprlemss1l  7257