Theorem recexprlemell 7278

Description: Membership in the lower cut of 𝐵. Lemma for recexpr 7294. (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 2644	. 2 ⊢ (𝐶 ∈ (1st ‘𝐵) → 𝐶 ∈ V)
2		ltrelnq 7021	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4519	. . . . . 6 ⊢ (𝐶 <Q 𝑦 → (𝐶 ∈ Q ∧ 𝑦 ∈ Q))
4	3	simpld 111	. . . . 5 ⊢ (𝐶 <Q 𝑦 → 𝐶 ∈ Q)
5		elex 2644	. . . . 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 1541	. 2 ⊢ (∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝐶 ∈ V)
9		breq1 3870	. . . . 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 1760	. . 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 5343	. . . 4 ⊢ (1st ‘𝐵) = (1st ‘⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩)
14		nqex 7019	. . . . . 6 ⊢ Q ∈ V
15	2	brel 4519	. . . . . . . . . 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 1541	. . . . . . 7 ⊢ (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
19	18	abssi 3111	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ⊆ Q
20	14, 19	ssexi 3998	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))} ∈ V
21	2	brel 4519	. . . . . . . . . 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 1541	. . . . . . 7 ⊢ (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 𝑥 ∈ Q)
25	24	abssi 3111	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ⊆ Q
26	14, 25	ssexi 3998	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))} ∈ V
27	20, 26	op1st 5955	. . . 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 2115	. . 3 ⊢ (1st ‘𝐵) = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}
29	11, 28	elab2g 2776	. 2 ⊢ (𝐶 ∈ V → (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))))
30	1, 8, 29	pm5.21nii 658	1 ⊢ (𝐶 ∈ (1st ‘𝐵) ↔ ∃𝑦(𝐶 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1296  ∃wex 1433   ∈ wcel 1445  {cab 2081  Vcvv 2633  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  1^st c1st 5947  2^nd c2nd 5948  Qcnq 6936  *_Qcrq 6940   <_Q cltq 6941

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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-qs 6338  df-ni 6960  df-nqqs 7004  df-ltnqqs 7009

This theorem is referenced by:  recexprlemm  7280  recexprlemopl  7281  recexprlemlol  7282  recexprlemdisj  7286  recexprlemloc  7287  recexprlem1ssl  7289  recexprlemss1l  7291